Higher-Spin Theories and theAdS/CFT Correspondence

Shailesh Lal

Enrolment Number : PHYS08200604019

Harish-Chandra Research Institute

A thesis submitted to the

Board of Studies in Physical Sciences Discipline

in partial fulfillment of requirements for the degree of

DOCTOR OF PHILOSOPHY

of

Homi Bhabha National Institute

August 2012

Certificate

This is to certify that the Ph. D. thesis titled “Higher-Spin Theories and the AdS/CFT Cor-respondence” submitted by Shailesh Lal is a record of bona fide research work done under mysupervision. It is further certified that the thesis represents independent work by the candidateand collaboration was necessitated by the nature and scope of the problems dealt with.

Date: Prof. Rajesh GopakumarThesis Advisor

Declaration

This thesis is a presentation of my original research work. Whenever contributions of others areinvolved, every effort is made to indicate this clearly, with due reference to the literature andacknowledgment of collaborative research and discussions.

The work is original and has not been submitted earlier as a whole or in part for a degree ordiploma at this or any other Institution or University.

This work was done under guidance of Professor Rajesh Gopakumar, at Harish-Chandra ResearchInstitute, Allahabad.

Date: Shailesh LalPh. D. Candidate

Acknowledgments

It is a pleasure to acknowledge the guidance and enouragement of my advisor, Rajesh Gopaku-mar, without which the work contained in this thesis could not have been carried out. I also thankhim for his many insightful comments and patient explanations which have strongly influenced andguided the work represented here, as well as the other problems I worked on during my time atHRI. It is also a pleasure to acknowledge the many helpful discussions I have had with membersof the string theory group at HRI, Ashoke Sen, Dileep Jatkar, Sudhakar Panda, SatchidanandaNaik, Anirban Basu, Justin David, Debashis Ghoshal, Yogesh Srivastava, Bobby Ezhuthachan,Chetan Gowdigere, Jae-Hyuk Oh, Akitsugu Miwa, Sayantani Bhattacharyya, Binata Panda, Su-vankar Dutta, Nabamita Banerjee, Shamik Banerjee, Ayan Mukhopadhyay, Ipsita Mandal, RojiPius, Swapnamay Mandal and Abhishek Chowdhury. I would especially like to thank Suvrat Raju,Rajesh Gupta, Arjun Bagchi, Arunabha Saha and Bindusar Sahoo for collaboration during thisperiod as well as several very stimulating discussions. I would also like to thank especially AshokeSen for supervising a reading course in string theory in my first year at HRI.

My work has also benefited greatly from many very helpful conversations and discussions.I would like to thank Hamid Afshar, Massimo Bianchi, Niklas Beisert, Andrea Campoleoni,Matthias Gaberdiel, Daniel Grumiller, Suresh Govindarajan, Jose Morales, Albrecht Klemm,Mikhail Vasiliev and Xi Yin for very helpful discussions.

I would also like to thank Pinaki Majumdar for supervising a very instructive project duringmy fourth semester at HRI and Venkat Pai, Viveka Nand Singh and Rajarshi Tiwari for severalvery enjoyable discussions on aspects of condensed matter physics. I would like to thank BiswarupMukhopadhyay and especially Nishita Desai for explaining many aspects of phenomenology andcollider physics and for many stimulating discussions.

More generally, I would like to thank the academic and the administrative wings of HRI forproviding a wonderful atmosphere to live and work in. I will certainly miss the many good timesI have had here. I thank, apart from people mentioned above, Jayanti, Pomita, Brundaban,Sudhir, Anamitra, Arijit, Akhilesh Nautiyal, Kalpataru, Subhaditya, Priyotosh, Archana, DheerajKulkarni, Satya, Dhiraj, Joydeep, Sanjoy, Jaban, Karam Deo, Bhavin, Pradip Misra, Pradeep Rai,Ram Lal and Vikas for sharing these good times with me. I would also like to thank Rukmini Deyfor organising several very enjoyable musical get-togethers.

I would also like to thank my teachers at the University of Delhi, particularly Amitabha Mukher-jee, Debajyoti Choudhury, Patrick Das Gupta, and Sudhendu Rai Chaudhuri for very helpful dis-cussions which have greatly benefited my overall understanding of physics. I would especially liketo thank Amitabha Mukherjee for supervising my M.Sc. dissertation. It is chiefly out of thisexperience that I developed the inclination to carry out research. I also thank him and DebajyotiChoudhury for encouragement to pursue research in theoretical physics.

I would like to thank here my family especially my mother, Manika, grandmother, Nila, andsister, Devika, for their constant support and encouragement. I also thank Atul, Mukul, Bipul,Mohit, Anant, Avijit, Savita, Abhishek, Aneesha, Raghav, Madhav, Vaibhav, Gauri, Ambika,Karn and Vanyaa for their support. I also remember here my father, Lalit, and my grandfathersB.B. Lal and B. S. Saxena who inspired me by personal example to pursue academic work and to“read, read and read!”. I would like to thank Pearl, Vijay, Satish, JJ, Damini, Dhruv, Hemant,Shishir, Anju, Radha, Kerry, Maggie and Tracy for their friendship. I thank Jaspal Singh forteaching me the classical guitar and introducing me to Francisco Tarrega’s wonderful music.

Finally, my work owes greatly to Oswald Summerton’s support and guidance and it is ultimatelyto him that I dedicate this thesis and the efforts that have gone into it.

Higher-Spin Theories and the AdS/CFT Correspondence

Thesis Synopsis

Shailesh LalHarish-Chandra Research Institute

—————————————————————-

The AdS/CFT Correspondence is a remarkable output of string theory—a theory of quantumgravity—which conjectures that a string theory defined on spacetimes that asymptote to Anti de-Sitter (AdS) spaces are actually completely equivalent to a conformal field theory (CFT) which isdefined on the spatial boundary of the AdS. Since a conformal field theory is a usual relativisticQuantum Field Theory, which we understand well, this gives us a tool to explore string theory andQuantum Gravity. However, though physicists have tested this duality and have always found it tobe true, many aspects of this duality remain unclear. For example, supersymmetry is ubiquitousin the known instances of this duality. At the same time, the general reasons for expecting a stringdescription of Quantum Field theories are quite independent of supersymmetry. We expect thatinsight into the mechanics of this duality will give us intuition for more ambitious questions likeformulating a string dual for Quantum Chromodynamics, the theory of strong interactions withinthe nucleus.

This thesis describes out work on the holography of higher-spin theories in AdS. The motivationsare two-fold: firstly, in the low-dimensional examples of AdS3 and AdS4, these theories are expectedto have CFT duals in their own right. These dualities are non-supersymmetric and therefore offera rare concrete opportunity to explore AdS/CFT in non-supersymmetric settings. Secondly, themajor thrust of research in AdS/CFT has been towards using the regime of classical weak gravity(Einstein’s General Relativity) as a tool to understand the physics of strongly coupled planargauge theories, which is typically intractable via standard field theory techniques. Alternatively,one might try to probe the opposite corner in the landscape of the duality, that of free planargauge theories, which correspond to tensionless string theory on AdS spacetimes. This is a naturalstarting point to get insight about the mechanics of AdS/CFT and gauge-string duality. However,even for supersymmetric settings, this limit of string theory has hitherto proven intractable. Herewe may expect that the higher-spin symmetry might play the role of a different large symmetrygroup that might render the problem tractable. As a subset of the full duality, we also expect thatin the tensionless limit, there is a sector of the string theory which consists of an infinite towerof massless, interacting particles, which should be described by a consistent classical theory. It isnatural to expect that this theory is a Vasiliev theory. Our work in higher-spin theories is withthese (not unrelated) questions in mind.

An important tool in our exploration of higher-spin theories will be the one-loop partitionfunction of the AdS higher-spin theory, which contains the leading quantum corrections to itsclassical description. We will define this partition function in terms of the heat kernel of theLaplacian acting on symmetric, transverse traceless tensors of arbitrary rank. Evaluating theheat kernel over AdS is a priori a hard problem, as it involves determining the eigenvalues andeigenfunctions of the Laplacian in curved spaces over tensors of arbitrary rank. A useful tool toexploit is the fact that AdS is a symmetric space, and these questions have a group theoreticinterpretation. A simple example well known from Quantum Mechanics is that the eigenfunctionsof the Laplacian acting on the wave function of a particle moving on the two-dimensional sphere arematrix elements of Wigner matrices, and the eigenvalues can be interpreted as quadratic Casimirsof SU(2). We use this fact to determine a closed-form expression the heat kernel of the Laplacian

for arbitrary spin fields over quotients of (Euclidean) AdS. This expression is in terms of charactersand quadratic Casimirs of unitary representations of the Lorentz group and the special orthogonalgroup, which have been completely determined in the existing literature.

Also, motivated by the Gaberdiel-Gopakumar conjecture relating higher-spin theories in AdS3

to WN minimal models, we consider a topologically massive deformation of these theories. Bythis, we mean that we consider a (parity violating) spin-s free field theory which has the linearisedhigher-spin gauge symmetry, but whose excitations include massive modes. This is nontrivial,because typically adding mass terms to gauge fields destroys the gauge symmetry. In particular,we compute the one-loop partition function of a topologically massive spin-3 field. This providesus with evidence that theW symmetry survives the chiral limit, which was a question unclear fromprevious analyses. We also compute the partition function for arbitrary spin fields, which enablesus to identify generalisations of the spin-1 excitation seen in the spin-3 field. A spin-s field containstransverse traceless excitations of spin s, s − 1, . . . , 1. We expect that these expressions will bean important ingredient in formulating an explicit duality for topologically massive higher-spintheories.

We finally turn to the computation of the partition function of higher-spin theories in AdS5. Weconsider the higher-spin theory containing all spins from zero to infinity appearing only once. Wefind simplifications in the multiparticle partition function. In particular, for thermal AdS5, we areable to write the answer in a form strongly suggestive of a vacuum character, and also in a formthat is reminiscitive of a higher-dimensional MacMahon function, generalising the observation ofGaberdiel, Gopakumar and Saha, who noted that the AdS3 partition function could be expressedin terms of the two-dimensional MacMahon function. We also turn on chemical potentials forthe other AdS5 Cartans and compute the one-loop partition function. We again obtain a vacuumcharacter-like form for the answer. We take this as evidence that the asymptotic symmetry algebraof higher-spin theories in AdS5 gets enhanced, in a manner similar to that of AdS3, to includeadditional generators that act on multi-particle states of the theory.

List of publications that form the thesis of the candidate

1. Partition Functions for Higher-Spin theories in AdS. Rajesh Kumar Gupta, Shailesh Lal.JHEP 1207 (2012) 071, [arXiv:1205.1130]

2. One loop partition function for Topologically Massive Higher Spin Gravity. Arjun Bagchi,Shailesh Lal, Arunabha Saha, Bindusar Sahoo.JHEP 1112 (2011) 068, [arXiv:1107.2063] [arXiv:1107.2063]

3. The Heat Kernel on AdS. Rajesh Gopakumar, Rajesh Kumar Gupta, Shailesh Lal.JHEP 1111 (2011) 010 [arXiv:1103.3627]

List of papers of the candidate that are not included in the thesis

1. Topologically Massive Higher Spin Gravity. Arjun Bagchi, Shailesh Lal, Arunabha Saha,Bindusar Sahoo.JHEP 1110 (2011) 150, [arXiv:1107.0915]

2. Rational Terms in Theories with Matter. Shailesh Lal, Suvrat Raju.JHEP 1008 (2010) 022, [arXiv:1003.5264]

3. The Next-to-Simplest Quantum Field Theories. Shailesh Lal, Suvrat Raju.Phys. Rev. D81 (2010) 105002, [arXiv:0910.0930]

Contents

I Introduction 1

1 Introduction 3

1.1 The AdS/CFT Correspondence: A review . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 The Geometry of AdSd+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Arriving at the AdS5/CFT4 duality . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Computations with the AdS/CFT Correspondence . . . . . . . . . . . . . . 7

1.1.4 Preliminary Checks on the AdS/CFT Correspondence . . . . . . . . . . . . 7

1.2 The Higher-Spin CFT Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 The Higher-Spin window of the AdS/CFT Correspondence . . . . . . . . . 10

1.3 Free Higher-Spin Theories in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

II Partition Functions for Higher-Spin Theories in AdS. 13

2 The Heat Kernel on AdS 15

2.1 The Heat Kernel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The Heat Kernel for the Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 The Heat Kernel on Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 The Heat Kernel on S2n+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 The Heat Kernel on Quotients of Symmetric Spaces . . . . . . . . . . . . . . . . . 20

2.4.1 The Thermal Quotient of S5 . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2 The Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.3 The Traced Heat Kernel on thermal S2n+1 . . . . . . . . . . . . . . . . . . 22

2.5 The Heat Kernel on AdS2n+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.2 Harmonic Analysis on H2n+1 . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.3 The coincident Heat Kernel on AdS2n+1 . . . . . . . . . . . . . . . . . . . . 25

2.6 The Heat Kernel on Thermal AdS2n+1 . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.1 The Thermal Quotient of AdS . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.2 The Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 The one-loop Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7.1 The scalar on AdS5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 An extension to Even Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 The Partition Function for Higher-Spin Theories in AdS 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 The Partition Function for a Higher-Spin Field in AdS . . . . . . . . . . . . . . . . 32

3.2.1 The Blind Partition Function in AdS5 . . . . . . . . . . . . . . . . . . . . . 33

3.2.2 Additional Chemical Potentials in AdS5 . . . . . . . . . . . . . . . . . . . . 34

3.3 The Refined Partition Function in AdS5 . . . . . . . . . . . . . . . . . . . . . . . . 35

i

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 The Partition Function for Topologically Massive Higher-Spin Theory 394.1 Three-Dimensional Gravity and its Topologically Massive deformation . . . . . . . 394.2 Higher-Spin Theories in AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 The basic set up for s = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.4 One-loop determinants for spin-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.5 Generalisation to arbitrary spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

III Conclusions 49

5 Conclusions and Outlook 51

IV Appendices 53

A Normalising the Heat Kernel on G/H 55

B The Plancherel Measure for STT tensors on HN 57

C Blind Partition functions for AdS4 and AdS7 59

D Computing the Refined Partition Function 61

E Classical analysis for generic spins 65

Bibliography 69

ii

Part I

Introduction

Chapter 1

Introduction

Describing how interactions take place in nature has been the single aim of science over the pastmany centuries. From the efforts of scientists over the past century or so, a paradigm has emerged,containing some key elements which any theory of nature must contain.

First of all, any theory that attempts to describe particles moving near the speed of lightmust incorporate Einstein’s Relativity. If the situation is such that gravitational effects can beignored (this is true when the particles are very light) then Special Relativity is sufficient, but ifgravitational interactions are significant, then we must incorporate General Relativity.

Secondly, a theory that attempts to describe physics at lengths smaller than the size of an atommust incorporate Quantum Mechanics, according to which, particles have a “dual” descriptionas waves smeared out over space. Although this seems bizarre at first sight, it has been testedrigorously by experiments for a hundred years, and has always been confirmed. We now stronglybelieve that Quantum Mechanics is an essential element of any theory that aims to describe Nature.

The task before us, as physicists, is therefore to invent a framework in which both Relativityand Quantum Mechanics may be incorporated. This is essentially the minimal prerequisite fora unified framework to describe all interactions that occur in nature. For Special Relativity andQuantum Mechanics, this program has been implemented in a mathematical framework known asQuantum Field Theory (QFT). This is the framework in which, for example, the quantum theoryof electricity and magnetism is formulated.

At the time of writing, there are several important questions which have been only partiallyaddressed or where only preliminary explorations have been made, largely due to the complexityof the problems. Here are some of them:

1. The interactions within the nucleus of an atom.

2. High temperature superconductivity, and other phenomena such as relativistic fluid flows.

3. A complete theory of Quantum Gravity, which we expect will be crucial to understandconcretely many questions about Black Holes and the Big Bang singularity.

String Theory is a theory of quantum gravity, and as such aims to solve 3. In the mid 1990sa remarkable conjecture was proposed by Maldacena [1], provides a novel interrelation betweenthe three questions outlined above. He conjectured that string theory (quantum gravity) definedon asymptotically AdS spaces is completely equivalent to a conformal field theory that lives onthe boundary of the AdS. By translating questions about quantum gravity to the quantum fieldtheory, we can address questions about quantum gravity without ever having to directly quantisegravity! Equally interestingly, by using the fact that General Relativity (or Supergravity) is anapproximate description of the underlying string theory, one can use this correspondence to shedlight on important questions about the hologram QFT. This conjecture is known as the AdS/CFT

3

correspondence (or AdS/CFT).

We shall soon see how higher-spin theories fit into this framework, but let us first review thecorrespondence itself.

1.1 The AdS/CFT Correspondence: A review

In this section, we shall briefly review some essential elements of the AdS/CFT duality. Firstly,we will recollect some kinematical elements that make such a duality possible. In particular, weshall show how the isometry group of AdS acts as the conformal group on its spatial boundary.A chief element of this dictionary is that the AdS hamiltonian gets identified to the dilatationoperator of the CFT. Secondly, we will then recapitulate a string theory argument from which thisduality arises. Thirdly, we shall then present some preliminary tests of this duality, which havesome non-trivial aspects in their own right.

1.1.1 The Geometry of AdSd+1

We begin with a brief review of the AdS geometry. This has been very extensively reviewed inthe literature, and we shall essentially follow [2, 3] here. One way of thinking about AdS space isto embed it in a higher-dimensional pseudo-Euclidean space. Let us consider a d+ 2 dimensionalspace M2,d and coordinates (y0,−→y , yd+1). The invariant length is

y2 = (y0)2 + (yd+1)2 −−→y 2, (1.1.1)

which is preserved by the group SO(2, d). Then AdSd+1 may be defined as the subspace

y2 = b2, (1.1.2)

where b is a constant. A convenient parametrisation of this space is

y0 = b cosh ρ cos τ, yd+1 = b cosh ρ sin τ, −→y = b sinh ρ−→Ω , (1.1.3)

where−→Ω are coordinates on Sd−1, 0 ≤ τ < 2π, and 0 ≤ ρ in order to cover the AdS once. Clearly,

SO(2, d) is the symmetry group of AdS, manifest from the embedding (1.1.2) into M2,d.

To access the boundary of AdS, we shall take the ys on the locus (1.1.2) to be very large, via ascaling

ya = Rua, (1.1.4)

where R is a real number which we shall take to infinity. In the coordinate system (1.1.3) thiscorresponds to taking ρ ' 0. In this limit, the locus (1.1.2) becomes

(u0)2 + (ud+1)2 = −→u 2, (1.1.5)

which naively looks d+1 dimensional. However, we could have scaled by any other large parametertR instead of R. Therefore, in the u parametrisation of the boundary, there is an equivalence

u 7→ tu, (1.1.6)

because of which the boundary is d dimensional. By fixing this arbitrariness in scaling, we canrepresent the boundary by

(u0)2 + (ud+1)2 = 1 = −→u 2. (1.1.7)

The boundary is therefore S1 ⊗ Sd−1. It is further possible to show that the AdSd+1 isometrygroup SO(2, d) acts on the boundary as the conformal group in d dimensions. We refer the readerto [2] for an explicit demonstration of this fact.

4

In this analysis we have so far omitted an important point. The induced metric on the AdSd+1

is given in the ‘global’ coordinates ρ, τ,Ω by

ds2 = b2(− cosh2 ρdτ2 + dρ2 + sinh2 ρdΩ2

). (1.1.8)

As it stands, since 0 ≤ τ < 2π, the geometry has closed timelike curves. To get around this,we unwrap the τ circle to let −∞ < τ < ∞. The boundary is then R ⊗ Sd−1. The rest of thestatements carry over as before. Especially, the SO(2) that corresponds to translations in theτ coordinate, i.e. the AdS Hamiltonian, gets mapped to dilatations in the boundary conformalgroup. In particular, there is a dictionary which relates the eigenvalues ∆ of operators in the CFTto the masses m of the fields in AdS that they are dual to, which we give some examples of.

• scalars: ∆± = 12

(d±√d2 + 4m2

),

• vectors: ∆± = 12

(d±

√(d− 2)2 + 4m2

),

• p-forms: ∆± = 12

(d±

√(d− 2p)2 + 4m2

),

• massless spin-2: ∆ = d.

We refer the reader to [3] and references therein for a more complete list.

For explicit computations, it is usually more convenient to work in the so-called Poincare patchof the AdS spacetime given by coordinates (z, t,−→x )

y0 =z

2

(1 +

1

z2

(b2 +−→x · −→x − t2

)), yd+1 =

bt

z,

yi =bxi

z, i = 1 . . . d− 1,

yd =z

2

(1− 1

z2

(b2 −−→x · −→x + t2

)).

(1.1.9)

with the metric

ds2 =1

z2

(−dt2 + dxidxi + dz2

). (1.1.10)

The boundary of the AdS is located at z = 0. These coordinates, however, cover only a part ofthe AdS spacetime.

As a final observation, we note that any geometry which is asymptotically AdS has an inducedconformal structure on its boundary. In particular, the asymptotic AdS Killing vectors act asconformal Killing vectors on the boundary. This leads to an important subtlety in identifying theAdSd+1 isometry group to the d dimensional conformal group for the special case of d = 2. Inparticular, the AdS isometry group is SL(2,R) ⊗ SL(2,R), which is just the global subgroup ofthe two-dimensional conformal group. How do we interpret the action of the other generators? Itturns out that the additional generators act as large gauge transformations on the AdS vacuum,generating multiparticle boundary graviton states from the vacuum [7].

1.1.2 Arriving at the AdS5/CFT4 duality

We have so far demonstrated how the AdS isometry group acts as the group of conformal transfor-mations on the boundary of AdS. This is still very far from arguing how a string theory defined onasymptotically AdS spacetimes may dynamically be the same as a CFT defined on its boundary.

We shall now review the central arguments of [1] from which this conjecture arises. This hasbeen extensively reviewed in the literature, see for example [2–5], and we shall only recapitulatethe central argument, closely following [5].

Our starting point will be Type IIB string theory in the supergravity approximation, in which

5

we shall consider the black 3-brane metric for N units of flux. We take N to be large throughout.In the string frame, this metric is given by

ds2 = H−12 (r) ηµνdx

µdxν +H12 (r) dxmdxm, (1.1.11)

where µ, ν range from 0 to 3 and parametrize the directions parallel to the brane, and m runs from4 to 9 and parametrizes the directions transverse to the brane, and we further have

H = 1 +L4

r4, L4 = 4πgNα′2, r2 = xmxm, (1.1.12)

where g is the string coupling. The horizon of this geometry is located at r = 0. In the nearhorizon limit, this metric becomes

ds2 =r2

L2ηµνdx

µdxν +L2

r2dr2 + L2dΩ2

5, (1.1.13)

which is the AdS5⊗S5 geometry. The curvatures of both the AdS and the sphere are determinedby the radius L. When gN >> 1, the radius L is large compared to the string length, andsupergravity is a good approximation to the string theory.

Parallely, we consider N D3 branes in the IIB string theory in flat space. Perturbation theoryabout this system is parametrized by gN , every world-sheet boundary on the branes brings a factorof N from the Chan-Paton trace and g from the genus. Clearly, perturbation theory is valid whengN << 1.

It turns out that that the D-branes of string theory and the branes of supergravity are com-plementary descriptions of the same system [6] . When gN is small, string perturbation theoryis valid, and the D-brane picture is appropriate, and when gN is large, the supergravity pictureshould be used.

The AdS/CFT duality arises from considering low-energy limits of both pictures. In the openstring desciption, this consists of massless closed and open strings, localised on the brane. Themassless closed string excitations are the supergravity multiplet, which decouples in the low-energylimit, and the open string excitations give the field theory defined on the world-volume of the D3branes, namely N = 4 super-Yang-Mills with gauge group SU(N).1

In the supergravity picture, valid when gN >> 1, we have massless closed strings away from thebrane, which decouple from the near-brane physics due to the infinite red-shift contained in g00

as r approaches 0. Near the brane, however, any massive closed string mode will have arbitrarilysmall energy, again due to the divergent red-shift.

Making the assumption that the low-energy limit commutes with varying g from a small to alarge value leads us to conjecture that

D = 4, N = 4 SU(N) SYM ≡ IIB String theory on AdS5 × S5. (1.1.14)

Let us consider how the parameters of string theory, the AdS radius L, the Planck length LP , andthe string tension α′−1, relate to the gauge theory parameters, gYM and N .

It turns out (see, for example, [5]) that upto overall proportionality constants,

1

g2∼(L

LP

)8

∼ N2,L

α′1/2∼(g2YMN

) 14 ≡ λ

14 , (1.1.15)

where λ is the ’t Hooft coupling. From this, it is clear that a planar gauge theory at a large value ofλ corresponds to a low-energy (large string tension) classical theory in the bulk i.e. supergravity.

1Naively, we would get U(N) as the gauge group, but the U(1) corresponds to the motion of the stack of D-branes,and decouples.

6

We shall shortly consider the ‘opposite’ high-energy (zero tension) classical bulk theory, whichcorresponds to λ = 0 in the boundary CFT.

1.1.3 Computations with the AdS/CFT Correspondence

We have so far presented an argument that string theory on AdS spacetimes is the same theoryas a conformal field theory defined on the boundary of the AdS. We now recollect the elementsof a precise dictionary between the two theories, i.e. we show how observables of one theory mapto observables of the other. This was identified by [11, 12]; and is reviewed, for example, in [3].We shall follow the treatment in section 3.1.2 of [3]. Essentially, to make the correspondencecomputable, one posits that the partition functions of the string theory in AdS is equal to thatof the CFT on the boundary. In particular, it is proposed that boundary values of AdS fields actas sources for operators in the field theory (this statement needs to be interpreted carefully, as weshall soon see). In particular, that

⟨e∫d4xφ0(x)O(x)

⟩CFT

= Zstring

[φ(x, z)|z=0 = φ0(x)

]. (1.1.16)

In particular, let us consider the simplest case of a massive scalar in AdSd+1 sourcing an operatorO in the CFTd. The wave equation for a massive scalar in AdS may be solved in the Poincarecoordinates (1.1.10) to obtain the asymptotic behaviour

φ(x, z) ∼ A(x) · zd−∆ +B(x) · z∆, (1.1.17)

where

∆ =d

2+

√d2

4+m2, (1.1.18)

as ∆ is typically greater that d. Now in the limit z → 0, the field does not take a definite value, itsbehaviour is dominated by zd−∆. We therefore modify (1.1.16) to impose the boundary condition

φ(x, ε) = εd−∆φ0(x). (1.1.19)

Since φ is an AdS scalar, φ0 must have length dimensions of ∆ − d, and hence the operator Omust have a conformal dimension ∆. This is precisely the correspondence mentioned previously.As a first application, one can use this prescription to compute, for example, two point functionsof the CFT as carried out by Witten in [12]. We will not pursue this course here.

1.1.4 Preliminary Checks on the AdS/CFT Correspondence

We have so far recapitulated the argument for string theory in asymptotically AdS spacetimes tobe dual to a conformal field theory on the boundary of the AdS. A very precise instance of thisconjecture is the duality between Type IIB string theory and N = 4 super-Yang-Mills with gaugegroup SU(N). This has a number of appealing features.

1. Firstly, it is a conjecture that a string theory, a theory of quantum gravity, which is pro-hibitively difficult to solve directly, is actually equivalent to a Yang-Mills theory, which ismuch easier to quantize. This gives us a framework to explore a quantum theory of gravitywithout having to directly quantize gravity (or string theory).

2. Secondly, it states – in effect – that a theory of quantum gravity in a given spacetime isequivalent to a theory defined on the boundary of the spacetime. It is natural to expect thatthe correspondence is holographic in the sense of [8, 9], and one can show [3] that this isindeed the case.

3. Finally, it has long been argued [10] that there is a string description of planar Yang-Mills

7

theories. This conjecture explicitly realises this expectation. The string description is how-ever in a higher number of spacetime dimensions than the Yang-Mills theory itself.

Thus we see that the AdS/CFT correspondence contains within itself explicit realizations of manyprofound ideas that have guided physicists for the past many years. A skeptic would perhaps bejustified in saying that this seems too good to be true, and it would certainly be required to subjectthis conjecture to as many tests as possible, pending a complete proof.

We shall now describe a few preliminary checks that can immediately be made. We begin withthe AdS5 example.

1. First, we note that AdS5⊗S5 has the isometry group SO(2, 4)⊗SO(6). This is precisely thegroup of (bosonic) symmetries of N = 4 super-Yang-Mills in four dimensions. The SO(2, 4)is the group of conformal transformations, and the SO(6) is the group of R-symmetries ofthe theory. This feature plays a key role in formulating a computational prescription for thiscorrespondence [11, 12].

2. Using the symmetry matching as a guideline, it is natural to propose that fields in AdS5

transforming under a given unitary irreducible representation of SO(2, 4) should correspondto gauge invariant operators transforming under the same representation of SO(2, 4) as re-viewed earlier in the section. Applying this to the case of short representations, we are ledto expect that boundary conserved currents should be dual to bulk massless fields. This isindeed the prescription of [11, 12].

3. It will soon become apparent that the window in which classical supergravity is a goodapproximation to the underlying string theory corresponds to the regime in which the dualgauge theory is a planar, strongly coupled theory. This raises a small puzzle (which isimmediately resolved). It is known that supergravity on AdS5 ⊗ S5 has infinite towers ofKaluza Klein modes arising from reduction on the five-sphere [13]. These are massive modesand do not saturate the SO(2, 4) unitarity bound, and yet their scaling dimension ∆ shouldbe protected from becoming very large in the strongly coupled gauge theory. It turns outthat these modes are dual to short representations of the full superconformal algebra, i.e.BPS states, in the dual field theory [12], and the conformal dimension of the dual states inthe QFT is therefore protected.

4. Finally, it is interesting to note that the parameter N is quantized on both sides of the theory,despite the fact that N was taken to be large in ‘deriving’ the duality. This may be takento indicate that the duality continues to be true at finite N .

Similar tests have also been carried out for other instances of the AdS/CFT correspondence.

1. In the AdS3 example, the spectrum matching leads to a non-trivial string exclusion principlefrom the conformal field theory. The existence of this principle is not manifest from thesupergravity in AdS3, but it may be shown explicitly that the spectrum indeed matches. Werefer the reader to [3] for details.

2. In the ABJ(M) dualities [14, 15] in AdS4, spectrum matching again has very non-trivialconsequences. In particular, this requires the introduction of new monopole operators in thedual gauge theory. This was a non-trivial prediction of the AdS/CFT correspondence, whichwas later confirmed by an explicit construction of these operators. We refer the reader tothe review [16] for details of these and related developments.

In addition, the correspondence has been successfully tested for the planar theory very extensivelyat arbitrary values of the ‘t-Hooft coupling using integrability (see the review [17]) and finally, thesedualities have also been put to very non-trivial tests at the non-planar level using localisation [18].These highly non-trivial tests are very convincing evidence for the validity of this duality.

8

1.2 The Higher-Spin CFT Correspondence

Despite the fact that no particle with spin greater than one has ever been experimentally detected,and gravitational waves – which would have helicity 2 – have yet to be seen, Higher-Spin theorieshave been explored with a great deal of interest for the past many years. This has notably resultedin the construction, by Vasiliev and collaborators, of classical theories in AdS involving an infinitenumber of massless particles of arbitrary spin. See [19] for a review of Vasiliev theories in generaldimensions.

There are a variety of reasons for this interest. First of all, there is a priori no reason whythere should be particles of spins only upto two. The Poincare group admits unitary irreduciblerepresentations of arbitrary (half-integer) spin. It is therefore interesting to explore if there couldbe consistent interacting theories with particles of spin higher than two. Also, the existence of con-served charges of spins higher than those of the Poincare group generators is explicitly forbidden bythe Coleman-Mandula theorem, and its supersymmetric extension, the Haag-Lopuszanski-Sohniustheorem [20]. Again, it would be interesting to see how this theorem might be evaded, and this isanother motivation to study higher-spin theories2. Also, from the form of equations of motion ofVasiliev’s higher-spin theories, there is an expectation that these will be in some way a controllabletoy model for String Field Theory.

In this thesis, however, we shall focus on the special relevance of higher-spin theories to theAdS/CFT Correspondence as the chief motivation for their exploration. The holography of higher-spin theories has been a topic of some sustained exploration [24]–[36]. This interest has essentiallybeen fuelled by two motivations.

Firstly, as we shall shortly review, it is reasonable to expect that the twist-two sector of a (free,planar) CFTd—such as N = 4 SYM—would be described in the AdSd+1 bulk by a consistentclassical theory with an infinite tower of arbitrary-spin particles. It is natural to expect that thisbulk theory would be a Vasiliev theory. See [37] for details. Since a free CFT is in some sense anatural starting point for organising the boundary theory into a string theory (see [38–40] for asystematic approach) probing these questions would be helpful in order to learn about the generalmechanics of gauge-string duality. Equally intriguingly, there is the additional possibility that evenwhen the boundary theory is not free, higher-spin symmetries are still present in a higgsed phase[41]. This, if true, would afford us insight into the larger set of gauge invariances of string theory.

Secondly, in the low-dimensional examples of AdS3 and AdS4, these theories are expected tohave CFT duals in their own right [24, 29, 31, 34, 35]. Typically, the best known and studiedexamples of AdS/CFT are supersymmetric [1], because supersymmetry allows us the freedom toreliably compute and extrapolate results on both sides of the duality. However, there are goodreasons to believe that the general phenomena of gauge-string duality and holography are nottied to supersymmetry. A natural allied question is to ask what role does supersymmetry playin AdS/CFT. In this regard, the holography of higher-spin theories is interesting because manyof these dualities are non-supersymmetric [24, 29, 32] and offer us an opportunity to concretelyexplore AdS/CFT away from supersymmetry.

The above statements require some qualification for the case of AdS3/CFT2 dualities like [29].In particular, the CFTs dual to higher-spin theories in AdS3 are not free, and in fact can even bestrongly coupled [42]. This is not surprising; the Coleman-Mandula theorem breaks down in twodimensions.

2In three dimensional CFTs, the recent Maldacena-Zhiboedov theorem [21] corresponds to the Coleman-Mandulatheorem. In particular, the existence of a single higher-spin conserved current can be shown to imply the existenceof infinitely many such conserved currents, and the correlation functions of the currents of the CFT equal thatof a free theory. The corresponding case, for arbitrary dimensions is interestingly more involved, and there is thepossibility of having additional structures in conserved currents, not arising from free field theory [22, 23]

9

1.2.1 The Higher-Spin window of the AdS/CFT Correspondence

To get a sense of how higher-spin theories can be related to the AdS/CFT duality let us considerthe AdS5/CFT4 duality, and the dictionary 1.1.15, which is the context in which the connectionof higher-spin theories to AdS/CFT originally arose [37, 43]. Similar statements would hold in allknown examples of the AdS/CFT duality.

We consider the window where gYM = 0 and N → ∞. This would correspond in the bulkto classical tensionless string theory (or equivalently, classical string theory formulated on a zeroradius AdS). In the boundary, this is a free, planar gauge theory with all matter in the adjointrepresentation (from supersymmetry requirements) and is in some sense, a natural starting pointfor organising a gauge theory into its string dual. See [38–40] for a general approach to thisproblem.

While the string theory in this limit is notoriously difficult to deal with – even with supersym-metry – we can use the boundary theory to develop an expectation of what the bulk will be like.This is essentially the approach adopted in [37, 43].

We begin by observing that a free theory contains an infinite number of conserved currents ofarbitrary spin. These have been constructed systematically in d dimensions, and we give the finalform only.

Jµ1...µs =s∑

k=0

(−1)k

k!(k + d−4

2

)!(s− k)!

(s− k + d−4

2

)!∂(µ1 . . . ∂µkφ

∗ (x) · ∂µk+1. . . ∂µs)φ (x)− traces,

(1.2.20)where the dot product is the invariant inner product defined over the representation of the gaugegroup that the fields φ transform in. For example, it would be a trace in the adjoint representation.It can further be shown that these operators form a closed subsector of the free conformal fieldtheory, in the sense that the OPE of these operators closes among themselves. We thereforeexpect that there is a subsector of the tensionless string theory in the bulk which is described by aconsistent, classical, interacting theory of an infinite number of particles of spins ranging from zeroto infinity. As mentioned previously, such theories are independently known of in AdS spacetimesand it seems natural to try and connect the conserved current subsector of free Yang-Mills theoriesto these higher-spin theories.

1.3 Free Higher-Spin Theories in AdS

We will begin with obtaining the quadratic action for free massless3 higher-spin fields when ex-panded about the AdS vacuum. By a spin-s field, we mean a symmetric rank-s tensor, subjectto constraints which we shall come to in a moment. It turns out that for (Anti-) de Sitter space-times, one can determine this action by taking the action for a massless spin-s field and minimallycoupling it to background gravity. This has been worked out in [44, 45]. We refer the reader to[28, 49] for a modern review, and [46–48] for recent progress. Free higher-spin theory in flat spaceis also reviewed in [50].

The main ingredient for the analyses carried out in this thesis is a quadratic action for a sym-metric rank-s tensor field, invariant under the gauge transformation

φµ1...µs 7→ φµ1...µs +∇(µ1ξµ2...µs). (1.3.21)

The brackets imply symmetrisation over the enclosed indices. This is achieved by adding theminimal number of terms necessary to get a completely symmetrised object. There is no over-allweight multiplied. For a consistent description, it turns out that the fields φ necessarily satisfy a

3The notion of masslessness for a tensor field is slightly ambiguous on a curved manifold such as AdS, by ‘massless’,we mean that the action has a gauge freedom, which we specify in (1.3.21).

10

double-tracelessness constraint which is trivial for spins 3 and lower.

φµ1...µs−4νρνρ = 0. (1.3.22)

The gauge transformation parameter ξ then satisfies a tracelessness constraint

ξµ1...µs−3νν = 0. (1.3.23)

Note that this constraint is non-trivial—and is imposed—even for the spin-3, which has no doubletracelessess constraint. With these conditions, a quadratic action for the spin-s field in AdSD,invariant under the gauge transformation (1.3.21) may be written down.

S[φ(s)

]=

∫dDx√gφµ1...µs

(Fµ1,...,µs −

1

2g(µ1µ2Fµ3...µs)λ

λ

), (1.3.24)

where

Fµ1...µs = Fµ1...µs −s2 + (D − 6) s− 2 (D − 3)

`2φµ1...µs −

2

`2g(µ1µ2φµ3...µs)λ

λ, (1.3.25)

and

Fµ1...µs = ∆φµ1...µs −∇(µ1∇λφµ2...µs)λ +

1

2∇(µ1∇µ2φµ3...µs)λ

λ. (1.3.26)

From this action, it already seems plausible that the result of evaluating the path integral in theone-loop approximation will be expressible in terms of the determinant of the Laplacian evaluatedover spin-s fields. It will soon become apparent that this is indeed the case. We shall thereforefocus on evaluating precisely these determinants in the next chapter.

11

Part II

Partition Functions for Higher-SpinTheories in AdS.

Chapter 2

The Heat Kernel on AdS

2.1 The Heat Kernel Method

As mentioned in the previous chapter, a complete theory of quantum gravity is a main outstandinggoal of physics. However, important information may be extracted about the quantum theory byperforming a loop expansion about a classical vacuum. In particular, it turns out that the leadingquantum properties may be extracted purely from classical data. This is clearly important fromthe point of view of quantum gravity as this behaviour is potentially tractable even when we don’tknow the underlying quantum theory. We refer the reader to [51] for more details. To see how thisworks, consider a path integral for a field φ(x).

Z [Φ] =

∫DΦe−

i~S[Φ]. (2.1.1)

We will evaluate this in the limit ~→ 0 by the method of steepest descents to incorporate the lead-ing quantum corrections. In this limit, the path integral is dominated by classical configurations,which solve

δ

δΦS [Φ] |Φ=Φcl = 0 (2.1.2)

Let us decomposeΦ = Φcl + φ, (2.1.3)

where φ denotes small (quantum) fluctuations about the classical configuration Φcl. In an expan-sion about Φcl, we may heuristically write

S [Φcl + φ] ' S [Φcl] +δ2

δΦ2S [Φ] |Φ=Φclφ

2. (2.1.4)

Or more correctly,

S = Scl + 〈φ,Dφ〉 where 〈φ1, φ2〉 =

∫dnx√gφ1(x)φ2(x). (2.1.5)

The linear term in the variation vanishes because Φcl obeys the classical equation of motion andhence extremises the action. We have omitted higher-order terms in this expansion which corre-spond to higher loop corrections to Z. The one-loop path integral is therefore a Gaussian integraland may be easily evaluated.

Z1−` [φ] = det−12 (D) . (2.1.6)

15

The operator D should at least be a self-adjoint operator, we will also assume that it is elliptic(so that its eigenvalues are of definite sign). There are additional technical assumptions for whichwe refer the reader to [51]. In this chapter, we will consider the case that D is the Laplacian andφ is an arbitrary spin field moving on an arbitrary-dimensional AdS background and explicitlyevaluate the determinant above by relating it to the trace of the exponential of the Laplacian.

As a final remark, if Φ is a gauge field then there is a gauge redundancy which has to be fixedwhile evaluating the path integral. Typically, this involves the introduction of ghost fields, whichyield determinants of their own. See for example, [52] for the corresponding Yang-Mills example.We will also explicitly evaluate these by these heat kernel methods.

2.2 The Heat Kernel for the Laplacian

Given the normalised eigenfunctions ψ(S)n,a (x) of the Laplacian ∆(S) for a spin-S field on a manifold

Md+1, and the spectrum of eigenvalues E(S)n , we can define the heat kernel between two points x

and y as

K(S)ab (x, y; t) = 〈y, b|et∆(S) |x, a〉 =

∑n

ψ(S)n,a (x)ψ

(S)n,b (y)∗ etE

(S)n , (2.2.7)

where a and b are local Lorentz indices for the field. We can trace over the spin and spacetimelabels to define the traced heat kernel as

K(S) (t) ≡ Tret∆(S) =

∫M

√gdd+1x

∑a

K(S)aa (x, x; t) . (2.2.8)

The one-loop partition function is related to the trace of the heat kernel through

lnZ(S) = ln det(−∆(S)

)= Tr ln

(−∆(S)

)= −

∫ ∞0

dt

tTret∆(S) . (2.2.9)

For a general manifold, the computation of the heat kernel is a formidable task, even for thescalar Laplacian. Typically one has asymptotic results. However, for symmetric spaces such asspheres and hyperbolic spaces (Euclidean AdS) there are many simplifications. This is becausethese spaces can be realised as cosets G/H and one can therefore use the powerful methods ofharmonic analysis on group manifolds. In [53] these techniques were used to explicitly computethe heat kernel on thermal AdS3 for fields of arbitrary spin. Though [53] exploited the group theory,it also used at several places the particular fact that S3 (from which one continued to AdS3) itselfis the group manifold SU(2). In fact, many properties of SU(2) were used in intermediate stepsand it was not completely obvious how these generalise to higher dimensional spheres or AdSspacetimes.

In this chapter, we will generalise the methods of [53] to compute the heat kernel for the Laplacianfor arbitrary spin tensor fields on thermal AdS spacetimes. This therefore includes the cases ofAdS4, AdS5 and AdS7 which play a central role in the AdS/CFT correspondence. Since we areprimarily interested in evaluating the one-loop partition function for a spin-S particle, we shallconcentrate on the traced heat kernel though we will see that the techniques are sufficiently general.One may, for instance, also adapt these techniques to evaluate other objects of interest such as the(bulk to bulk) propagator.

We shall mostly focus on M = AdS2n+1/Γ where Γ is the thermal quotient. For practicalreasons we will obtain the answer for thermal AdS by analytic continuation of the answer for anappropriate “thermal quotient” of the N -sphere. As explained in [53] there are good reasons tobelieve that this analytic continuation of the harmonic analysis works for odd dimensional spheresand hyperboloids (Euclidean AdS). For even dimensional hyperboloids there are some additionaldiscrete representations other than the continuous ones one obtains by a straightforward analytic

16

continuation. However, this additional set of representations does not contribute for Laplaciansover a wide class of tensor fields, which in particular include the symmetric transverse traceless(STT) tensors. We provide necessary details in Section 2.8.

We shall begin with a brief review of harmonic analysis on homogeneous spaces, which is themainstay of these computations.

2.3 The Heat Kernel on Homogeneous Spaces

The heat kernel of the spin-S Laplacian1 may be evaluated over the spacetime manifold M bysolving the appropriate heat equation. Alternatively, one may attempt a direct evaluation byconstructing the eigenvalues and eigenfunctions of the spin-S Laplacian and carrying out the sumover n that appears in (2.2.7). Both these methods quickly become forbidding when applied to anarbitrary spin-S field. However, ifM is a homogeneous space G/H, then the use of group-theoretictechniques greatly simplifies the evaluation. The main simplifications arise from the following factswhich we will review below and then heavily utilise:

1. The eigenvalues E(S)n of the Laplacian ∆(S) are determined in terms of the quadratic Casimirs

of the symmetry group G and the isotropy subgroup H. There is thus a large degeneracy ofeigenvalues.

2. The eigenfunctions ψ(S)n,a(x) are matrix elements of unitary representation matrices of G.

3. This enables one to carry out the sum over degenerate eigenstates using the group multi-plication properties of the matrix elements. Thus a large part of the sum in (2.2.7) can beexplicitly carried out.

We begin with a brief recollection of some basic facts about harmonic analysis on coset spaces.This will collate the necessary tools with which we evaluate (2.2.7) and (2.2.8), and will set upour notation. The interested reader is referred to [54–56] for introduction and details and to [53]for explicit examples of these constructions. Given compact Lie groups G and H, where H is asubgroup of G, the coset space G/H is constructed through the right action of H on elements ofG

G/H = gH. (2.3.10)

(We will also need to consider left cosets, Γ\G, where Γ will act on elements of G from the left.)

We recall that G is the principal bundle over G/H with fibre isomorphic to H. Let π be theprojection map from G to G/H, i.e.

π (g) = gH ∀g ∈ G. (2.3.11)

Then a section σ (x) in the principal bundle is a map

σ : G/H 7→ G, such that π σ = e, (2.3.12)

where e is the identity element in G, and x are coordinates in G/H. A class of sections which willbe useful later is of the form

σ (gH) = g, (2.3.13)

where g is an element of the coset gH, which is chosen by some well-defined prescription. Theso called ‘thermal section’ that we choose in Section 2.4.1 is precisely of this form. Let us labelrepresentations ofG byR and representations ofH by S. We will sometimes refer to representations

1By spin-S we refer to the representation under which the field transforms under tangent space rotations. In thecase of S2n+1 or (euclidean) AdS2n+1, this will be a representation of SO(2n + 1). The Laplacian is that ofa tensor field transforming in this representation. In the case of spheres and hyberboloids we will consider theLaplacian with the Christoffel connection.

17

S of H as spin-S representations.2 The vector space which carries the representation R is calledVR, and has dimension dR, while the corresponding vector space for the representation S is VS ofdimension dS .

Eigenfunctions of the spin-S Laplacian are then given by the matrix elements

ψ(S)Ia (x) = U (R)

(σ (x)−1

)a

I, (2.3.14)

where S is the unitary irreducible representation of H under which our field transforms, and R isany representation of G that contains S when restricted to H. a is an index in the subspace VS ofVR, while I is an index in the full vector space VR. Generally, a given representation S can appearmore than once in R. However, we shall be interested in the coset spaces SO (N + 1) /SO (N) andSO (N, 1) /SO (N), for which a representation S appears at most once [57, 58]. We have thereforedropped a degeneracy factor associated with the index a, which appears in the more completeformulae given in [58].

The corresponding eigenvalues are given by

− E(S)R,I = C2 (R)− C2 (S) . (2.3.15)

The index n for the eigenvalues of the spin-S Laplacian that appeared in (2.2.7) is therefore a pairof labels, viz. (R, I),3 where the eigenfunctions that have the same label R but a different I arenecessarily degenerate. We will therefore drop the subscript I for E(S).

The expression (2.2.7) for the heat kernel then reduces to

K(S)ab (x, y; t) =

∑R,I

a(S)R ψ

(S)(R,I),a (x)ψ

(S)(R,I),b (y)∗ etE

(S)R , (2.3.16)

where a(S)R = dR

dS1

VG/His a normalisation constant (see Appendix A). This can be further simplified

by putting in the expression (2.3.14) for the eigenfunctions.

K(S)ab (x, y; t) =

∑R

dR∑I=1

a(S)R U

(R)(σ (x)−1

)a

I[U (R)

(σ (y)−1

)b

I]∗etE

(S)R

=∑R

a(S)R U

(R)(σ (x)−1 σ (y)

)a

betE

(S)R , (2.3.17)

where we have used the fact that the U (R) furnish a unitary representation of G. As an aside,we note that this matrix representation of the group composition law is the generalisation of theaddition theorem for spherical harmonics on S2 to arbitrary homogeneous vector bundles on cosetspaces.

To establish notation for later use, we define the heat kernel with traced spin indices

K(S) (x, y; t) ≡dS∑a=1

K(S)aa (x, y; t) =

∑R

a(S)R TrS

(U (R)

(σ (x)−1 σ (y)

))etE

(S)R , (2.3.18)

where the symbol

TrS (U) ≡dS∑a=1

〈a, S|U|a, S〉, (2.3.19)

2In the case of sphere and hyperboloids, H is isomorphic to the group of tangent space rotations for the manifoldG/H. See footnote 1.

3Note that a labels the components of the eigenfunction and is not a part of the index n.

18

and can be thought of as a trace over the subspace VS of VR. Note that this restricted trace isinvariant under a unitary change of basis of VS and not invariant under the most general unitarychange of basis in VR.

2.3.1 The Heat Kernel on S2n+1

As a prelude to evaluating the traced heat kernel on the “thermal quotient” of the odd-dimensionalsphere, let us evaluate (2.3.17) for the case without any quotient. That is, we focus first onS2n+1 ' SO(2n+ 2)/SO(2n+ 1). We will describe the eigenfunctions (2.3.14) and define the sumover R explicitly. This will be useful when we analytically continue our results to the correspondinghyberbolic space. We begin by recalling some facts from the representation theory of specialorthogonal groups.

Unitary irreducible representations of SO(2n+ 2) are characterised by a highest weight, whichcan be expressed in the orthogonal basis as the array

R = (m1,m2, . . . ,mn,mn+1) , m1 ≥ m2 ≥ . . . ≥ mn ≥ |mn+1| ≥ 0 (2.3.20)

where the m1 . . .mn+1 are all (half-)integers. Similarly, unitary irreducible representations ofSO (2n+ 1) are characterised by the array

S = (s1, s2, . . . , sn) , s1 ≥ s2 ≥ . . . ≥ sn ≥ 0, (2.3.21)

where the s1 . . . sn are all (half-)integers.

Then the quadratic Casimirs for the unitary irreducible representations for an orthogonal groupof rank n+ 1 can be expressed as (see e.g.[57, 59]).

C2 (m1, . . . ,mn+1) = m2 + 2r ·m. (2.3.22)

Here the dot product is the usual euclidean one, and the Weyl vector r is given by

ri =

n− i+ 1 if G = SO (2n + 2) ,(n+ 1

2

)− i if G = SO (2n + 1) ,

(2.3.23)

where i runs from 1 to n+ 1.

Let us now consider the expression (2.3.17) for the spin-S Laplacian on S2n+1. The eigenvalues

E(S)R are given by (2.3.15) and (2.3.22) and may be written down compactly as

− E(S)R = m2 + 2rSO(2n+2) ·m− s2 − 2rSO(2n+1) · s. (2.3.24)

The corresponding eigenfunctions are given by (2.3.14) where we have to specify which are therepresentations R of SO (2n+ 2) that contain a given representation S of SO (2n+ 1). This isdetermined by the branching rules, which for our case state that a representation R given by(2.3.20) contains the representation S if

m1 ≥ s1 ≥ m2 ≥ s2 ≥ . . . ≥ mn ≥ sn ≥ |mn+1|, (mi − si) ∈ Z. (2.3.25)

Using these branching rules, one can show that the expression that appears on the right of (2.3.24)is indeed positive definite, so that the eigenvalue itself is negative definite as per our conventions.

These rules further simplify if we restrict ourselves to symmetric transverse traceless (STT)representations of H. These tensors of rank s correspond to the highest weight (s, 0, . . . , 0). Inthis case, some of these inequalities get saturated, and one obtains the branching rule

m1 ≥ s = m2 ≥ 0, (2.3.26)

19

with all other mi, si zero, 4 and the equality follows from requiring that R contain S in the maximalpossible way. Essentially, this is equivalent to the transversality condition. The sum over R thatappears in (2.3.17) is now a sum over the admissible values of m1 in the above inequality. Thusif we restrict ourselves to evaluating the heat kernel for STT tensors, then we will be left with asum over m1 only.

The expression (2.3.24) for the eigenvalue also simplifies in this case. With the benefit ofhindsight, we will write this in a form that is suitable for analytic continuation to AdS

− E(S)R = (m1 + n)2 − s− n2. (2.3.27)

Using these tools, one can write down a formal expression for the heat kernel for a spin-S particlebetween an arbitrary pair of points x and y on S2n+1 using (2.3.17). This is given by

KSab (x, y; t) =

∑mi

n!

2πn+1

dRdSU (R)

(σ (x)−1 σ (y)

)a

betE

(S)R , (2.3.28)

where we have simply expanded out the sum over R into a sum over the permissible values ofmi determined by (2.3.25), and inserted the expression for the volume of the (2n+ 1)-sphere.Expressions for the dimensions dR and dS are well known (see for example[57, 59]). We list themhere for the reader’s convenience.

dR =n+1∏i<j=1

l2i − l2jµ2i − µ2

j

, dS =n∏

i<j=1

l2i − l2jµ2i − µ2

j

n∏i=1

liµi, (2.3.29)

where li = mi + (n+ 1)− i, µi = (n+ 1)− i, li = si + n− i+ 12 , and µi = n− i+ 1

2 . As explainedabove, this expression further simplifies for the STT tensors, and we obtain

KSab (x, y; t) =

∑m1

n!

2πn+1

d(m1,s)

dsU (m1,s)

(σ (x)−1 σ (y)

)a

betE

(S)R , (2.3.30)

where the labels (m1, s) and s that appear on the RHS are shorthand for R = (m1, s, 0 . . . , 0) andS = (s, 0 . . . , 0) respectively. This expression should be compared with the equation (3.9) obtainedin [53] for the case of 3 dimensions. The traced heat kernel for the STT tensors is then given by

KS (x, y; t) =∑m1

n!

2πn+1

d(m1,s)

dsTrS

(U (m1,s)

(σ (x)−1 σ (y)

))etE

(S)R . (2.3.31)

As mentioned earlier, we can in principle use these formulae to construct explicit expressions forthe heat kernel between two points a la [53], and thus for the bulk to bulk propagator. This would,however also require using explicit matrix elements of SO (2n+ 2) representations, and we shallnot pursue this direction further here.

2.4 The Heat Kernel on Quotients of Symmetric Spaces

We consider the heat kernel on the quotient spaces Γ\G/H where Γ is a discrete group whichcan be embedded in G. Though it is not essential to our analysis, we will assume that Γ is offinite order and is generated by a single element. This is indeed true for the quotients (2.4.34)we consider here. In particular, for the “thermal quotient” on the N -sphere, Γ is isomorphic toZN . To evaluate the heat kernel (2.2.7) on this space, a choice of section that is compatible withthe quotienting by Γ is useful. By this we mean that if γ ∈ Γ acts on points x = gH ∈ G/H by

4For the special case of n = 1, i.e. S3, the branching rule is m1 ≥ s = |m2| ≥ 0.

20

γ : gH 7→ γ · gH, then a section σ (x) is said to be compatible with the quotienting Γ iff

σ (γ (x)) = γ · σ (x) . (2.4.32)

The utility of this choice of section will become clear when we explicitly evaluate the traced heatkernel (2.2.8) for such geometries.

2.4.1 The Thermal Quotient of S5

As an example, we consider the thermal quotient of S5. This will serve as a useful prototype tokeep in mind. We shall also see that we can extrapolate the analysis to the general odd-dimensionalsphere. To begin with, let us express the thermal quotient in terms of “triple-polar” coordinateson S5, which are complex numbers (z1, z2, z3) such that

|z1|2 + |z2|2 + |z3|2 = 1. (2.4.33)

We consider the quotientγ : φi 7→ φi + αi. (2.4.34)

Here φ1, φ2, φ3 are the phases of the z’s and niαi = 2π for some ni ∈ Z and not all nis aresimultaneously zero.5 However, to embed Γ in SO (6), it is more useful to decompose thesecomplex numbers into 6 real coordinates that embed the S5 into R6,

x1 = cos θ cosφ1 x2 = cos θ sinφ1,

x3 = sin θ cosψ cosφ2 x4 = sin θ cosψ sinφ2,

x5 = sin θ sinψ cosφ3 x6 = sin θ sinψ sinφ3. (2.4.35)

Now, we construct a coset representative in SO (6) for this point x with coordinates as in (2.4.35).To do so, we start with the point (1, 0, 0, 0, 0, 0) in R6, and construct a matrix g (x) that rotates thispoint, the north pole, to the generic point x. By construction, g (x) ∈ SO (6), and there is a one-to-one correspondence between the points x on S5 and matrices g (x), upto a right multiplicationby an element of the SO (5) which leaves the north pole invariant. Such a representative matrixg (x) can be taken to be

g (x) = eiφ1Q12eiφ2Q34eiφ3Q56eiψQ35eiθQ13 , (2.4.36)

where Q’s are the generators of SO (6). This is clearly an instance of a section in G over G/H.

The action of the thermal quotient (2.4.34) on the coset representative g (x) is

γ : g (x) 7→ g (γ (x)) = eiα1Q12eiα2Q34eiα3Q56 · x = γ · g (x) , (2.4.37)

where the composition ‘·’ is the usual matrix multiplication.6 This section has the property (2.4.32)that we demand from a thermal section. Hence, we choose the thermal section to be

σth (x) = g (x) . (2.4.38)

This analysis can be repeated for any odd-dimensional sphere to find the same expression for thethermal section. Essentially the only difference is that for a (2n+ 1)-dimensional sphere, we needto consider (n + 1) complex numbers zi and proceed exactly as above. We find that the thermalsection can always be chosen to be (2.4.38).

5Note that this is a more general identification than the thermal quotient we will need, where one can takeαi = 0 (∀i 6= 1)

6This gives the embedding of Γ in SO (6).

21

2.4.2 The Method of Images

Since the heat kernel obeys a linear differential equation- the heat equation- we can use the methodof images to construct the heat kernel on Γ\G/H from that on G/H (see, for example [60]). Therelation between the two heat kernels is

K(S)Γ (x, y; t) =

∑γ∈Γ

K(S) (x, γ (y) ; t) , (2.4.39)

where K(S)Γ is the heat kernel between two points x and y on Γ\G/H, K(S) is the heat kernel on

G/H and the spin indices have been suppressed. We shall use this relation to determine the tracedheat kernel on the thermal quotient of S2n+1.

2.4.3 The Traced Heat Kernel on thermal S2n+1

We use the formalism developed above to evaluate the traced heat kernel for the thermal quotientof the odd-dimensional sphere. Using the method of images, the quantity of interest is

K(S)Γ (t) =

∑k∈ZN

∫Γ\G/H

dµ (x)∑a

K(S)aa

(x, γk (x) ; t

), (2.4.40)

where dµ(x) is the measure on Γ\G/H obtained from the Haar measure on G, and x labels pointsin Γ\G/H. We have also used the fact that Γ ' ZN , and since the sum over m is a finite sum, theintegral has also been taken through the sum. Further, since∑

a

K(S)aa

(γx, γk (γx) ; t

)=∑a

K(S)aa

(x, γk (x) ; t

), (2.4.41)

the integral over Γ\G/H can be traded in for the integral over G/H. We therefore multiply by anoverall volume factor, and evaluate∫

G/Hdµ (x)

∑a

K(S)aa

(x, γk (x) ; t

), (2.4.42)

where dµ(x) is the left invariant measure on G/H obtained from the Haar measure on G, andx now labels points in the full coset space G/H. Putting the expression (2.3.18) into this, andchoosing the section (2.4.38) we obtain∫

G/Hdµ (x)

∑a

K(S)aa

(x, γk (x) ; t

)=

∫G/H

dµ (x)∑R

a(S)R TrS

(g−1x γkgx

)(R)etE

(S)R , (2.4.43)

where(g−1x γkgx

)(R)is an abbreviation for U (R)

(g (x)−1 γkg (x)

). As this expression stands, the

trace is only over some subspace VS ⊂ VR so the cyclic property of the trace cannot be used toannihilate the g−1

x with the gx. To proceed further, we move the integral into the summation toobtain∫

G/Hdµ (x)

∑a

K(S)aa

(x, γk (x) ; t

)=∑R

a(S)R

∫G/H

dµ (x) TrS

(g−1x γkgx

)(R)etE

(S)R . (2.4.44)

Since G and H are compact, we may use the property that [61]∫Gdgf (g) =

∫G/H

dµ(x)

[∫Hdhf (xh)

], (2.4.45)

22

where dg is the Haar measure on G, dµ and dh are the invariant measures on G/H and Hrespectively. x is an arbitrary choice of coset representatives that we make to label points in G/H.In what follows we shall choose the coset representative to be gx. Let us consider the function

f (g) = TrS

(g−1γkg

)(R). (2.4.46)

This function has the property that f (gxh) = f (gx). Putting this in (2.4.45), we see that theintegral over H becomes trivial, and we get∫

G/Hdµ (x) TrS

(g−1x γkgx

)(R)=

1

VH

∫GdgTrS

(g−1γkg

)(R). (2.4.47)

Now, as in [53], we note that the integral IG =∫G dg

(g−1γkg

)commutes with all group elements

g ∈ G because

IG · g =

∫Gdg(g−1γkg

)· g = g ·

∫Gd (gg)

((gg)−1 γk (gg)

)= g · IG, (2.4.48)

where we have used the right invariance of the measure, viz. dg = d (gg). We therefore have, fromSchur’s lemma, that

IG =

∫Gdg(g−1γkg

)∝ I, (2.4.49)

from which we obtain that∫GdgTrS

(g−1γkg

)(R)=dSdR

∫GdgTrR

(g−1γkg

)(R)=dSdRVGTrR

(γk). (2.4.50)

The quotient γ is just the exponential of the Cartan generators of SO (2n+ 2) (see, for example(2.4.37) for the S5). This trace, therefore is just the SO (2n+ 2) character χR in the representationR. Putting this result in (2.4.47), we find that∫

G/Hdµ (x) TrS

(g−1x γkgx

)(R)= VG/H

dSdRχR

(γk), (2.4.51)

where we have normalised volumes so that VG = VG/HVH . Therefore,

∑k∈ZN

∫G/H

dµ (x)K(S)(x, γk (x) ; t

)=∑k∈ZN

∑R

χR

(γk)etE

(S)R , (2.4.52)

where we have inserted the value of the normalisation constant a(S)R from (A.0.3). Now for the

thermal quotient, we have γ such that

α1 6= 0, αi = 0, ∀ i = 2, . . . , n+ 1. (2.4.53)

The volume factor for this quotient is just α12π . This gives us the traced heat kernel on the thermal

S2n+1

K(S)Γ (t) =

α1

2π

∑k∈ZN

∑R

χR

(γk)etE

(S)R . (2.4.54)

We find that the answer assembles naturally into a sum of characters, in various representationsof G, of elements of the quotient group Γ. Here the representations R of SO(2n + 2) are thosewhich contain S when restricted to SO(2n+ 1). The reader should compare this expression to theequation (4.20) obtained in [53].

23

2.5 The Heat Kernel on AdS2n+1

We have so far obtained an expression for the heat kernel on a compact symmetric space (2.3.17)and have extended our analysis to its left quotients. In particular, we have shown how the tracedheat kernel on (a class of) quotients of S2n+1 assembles into a sum over characters of the orbifoldgroup Γ. We now extend our analysis to the hyperbolic space H2n+1. Following the analysis of[53], we will use the fact that the N -dimensional sphere admits an analytic continuation to thecorresponding euclidean AdS geometry. We now give an account of how one can exploit this factto determine the heat kernel on AdS2n+1 and its quotients.

2.5.1 Preliminaries

Euclidean AdS is the N dimensional hyperbolic space H+N which is the coset space

HN ' SO (N, 1) /SO (N) , (2.5.55)

where the quotienting is done, as in the sphere, by the right action. As was done for the three-dimensional case, we will view SO (N, 1) as an analytic continuation of SO (N + 1).

For explicit expressions, we shall employ the generalisation of the triple-polar coordinates thatwe introduced in (2.4.35) to the general S2n+1. In these coordinates, the S2n+1 metric is

dθ2 + cos2θdφ21 + sin2θ dΩ2

2n−1. (2.5.56)

We perform the analytic continuation

θ 7→ −iρ, φ1 7→ it, (2.5.57)

where ρ and t take values in R, to obtain

ds2 = −(dρ2 + cosh2ρdt2 + sinh2ρ dΩ2

2n−1

). (2.5.58)

The reader will recognize this as the metric on global AdS2n+1, upto a sign.

Now, to construct eigenfunctions on HN , we need to write down a section in SO(N, 1). TheLie algebra of SO(N, 1) is an analytic continuation of SO(N + 1) where we choose a particularaxis- say ‘1’- as the time direction and perform the analytic continuation Q1j → iQ1j to obtainthe so(N, 1) algebra from the so(N + 1) algebra. This is equivalent to the analytic continuationof the coordinates described above. Therefore, the section in SO(N, 1) can be obtained from thatin SO (N + 1) by analytically continuing the coordinates via (2.5.57).

2.5.2 Harmonic Analysis on H2n+1

We have recollected basic results from harmonic analysis on coset spaces in Section 2.3, whichwe have exploited for compact groups G and H. In fact, all the basic ingredients that we haveemployed in our analysis can be carried over to the case of non-compact groups as well. Theeigenvalues of the spin-S Laplacian are still given by (2.3.15), and the eigenfunctions are still(2.3.14), i.e. they are determined by matrix elements of unitary representations of G. Theseunitary representations are now infinite dimensional, given that G is non-compact. However, forSO(N, 1), these representations have been classified [58, 62], and we shall use these results todetermine the traced heat kernel on AdS2n+1.

The only unitary representations of SO(N, 1) that are relevant to us are those that containunitary representations of SO(N). For odd-dimensional hyperboloids, where N = 2n + 1, theseare just the so-called principal series representations of SO (2n+ 1, 1) which are labelled by the

24

arrayR = (iλ,m2,m3, · · · ,mn+1) , λ ∈ R, m2 ≥ m3 ≥ · · · ≥ mn ≥ |mn+1|, (2.5.59)

where the m2, · · · ,mn and |mn+1| are non-negative (half-)integers. We shall usually denote thearray (m2,m3, . . . ,mn+1) by ~m. We also note that the principal series representations (iλ, ~m) thatcontain a representation S of SO (2n+ 1) are determined by the branching rules [58, 63]

s1 ≥ m2 ≥ s2 ≥ . . . ≥ mn ≥ sn ≥ |mn+1|, (2.5.60)

which, for the STT tensors just reduce to m2 = s with all other mis and sis set to zero (cf. 2.3.26),except for the case of n = 1, where the branching rule is |m2| = s. Comparing (2.5.59) to (2.3.20)suggests that the appropriate analytic continuation is

m1 7→ iλ− n, λ ∈ R+, (2.5.61)

which is indeed the analytic continuation used by [64]. Let us consider how the eigenvalues E(S)R

transform under this analytic continuation. It turns out that the eigenvalues (2.3.24) get continuedto

E(S)R,AdS2n+1

= −(λ2 + ζ

), ζ ≡ C2 (S)− C2 (~m) + n2, (2.5.62)

and that the eigenvalue for the STT tensors (2.3.27) gets continued to

E(S)R,AdS2n+1

= −(λ2 + s+ n2

). (2.5.63)

The eigenvalues on AdS have an extra minus sign apart from what is obtained by the analyticcontinuation because the metric S2n+1 under the analytic continuation goes to minus of the metriconAdS2n+1. This analytic continuation preserves the corresponding energy eigenvalue as a negativedefinite real number, which it must, because the Laplacian on Euclidean AdS is an elliptic operator,and its eigenvalues must be of definite sign.

2.5.3 The coincident Heat Kernel on AdS2n+1

In computing the heat kernel over AdS2n+1 by analytically continuing from S2n+1, the sum overm1 that entered in (2.3.28), (2.3.30) and (2.3.31) is now continued to an integral over λ. Ingeneral this integral over λ is hard to perform, but it simplifies significantly in the coincidentlimit and is evaluated below for this case. The traced heat kernel for STT tensors has previouslybeen obtained directly in this limit by [64] and this calculation therefore serves as a check of theprescription (2.5.57) of analytic continuation that we have employed. We will also see that thenormalisation constant aSR that appeared for the S2n+1 gets continued to µSR, which is essentiallythe measure for this integral. This is a brief summary of the calculation, the reader will find moredetails in Appendix B.

On using (2.3.29) for the special case of R = (m, s, 0, . . . , 0), one can show that aSR gets continuedvia (2.5.57) to µSR, where

µSR =1

ds

[λ2 +

(s+ N−3

2

)2]∏N−52

j=0

(λ2 + j2

)2N−1π

N2 Γ(N2

) (2s+N − 3) (s+N − 4)!

s! (N − 3)!, (2.5.64)

where N = 2n+ 1. A little algebra reveals this as the combination

µSR =CNg (s)

dS

µ (λ)

ΩN−1, (2.5.65)

in the notation of [64], (see their expressions 2.7 to 2.13 and 2.108 or our Appendix B). We

have omitted the overall sign (−1)N−1

2 in writing the above. The quantity µ (λ) is known as the

25

Plancherel measure. We now consider the expression (2.3.31) on S2n+1, which in the coincidentlimit, reduces to

KS (x, x; t) =∑m1

aSRdSetE

(S)R . (2.5.66)

where aSRdS = n!2πn+1d(m1,s). We analytically continue this expression via our prescription (2.5.57)

to the coincident heat kernel on AdS2n+1.

KS (x, x; t) =

∫dλµSR (λ) dSe

tE(S)R =

CNΩN−1

g (s)

∫dλµ (λ) e−t(λ

2+s+n2), (2.5.67)

which is precisely the expression obtained by [64].

2.6 The Heat Kernel on Thermal AdS2n+1

2.6.1 The Thermal Quotient of AdS

We are now in a position to calculate the traced heat kernel of an arbitrary tensor particle onthermal AdS2n+1. This space is the hyperbolic space H2n+1 with a specific Z identification (in thegeneralised polar coordinates)

t ∼ t+ β, β = iα1 (2.6.68)

which is just the analytic continuation by (2.5.57) of the identification (2.4.53) on the sphere. Sincet is the global time coordinate, β is to be interpreted as the inverse temperature.

2.6.2 The Heat Kernel

In section 2.5 we have discussed how the heat kernel on H2n+1 can be calculated by analyticallycontinuing the harmonic analysis on S2n+1 to H2n+1. As discussed in Sec. 6.2 of [53], we expectto be able to continue the expressions for the heat kernel on the thermal sphere to thermal AdS,with the difference that now Γ ' Z, rather than ZN . Also, as noted in [53], essentially the onlydifference that arises for the traced heat kernel is that the character of SO(2n+ 2) that appears in(2.4.54) is now replaced by the Harish-Chandra (or global) character for the non-compact groupSO(2n+ 1, 1).

With these inputs, the traced heat kernel on thermal AdS2n+1 is given by

K(S) (γ, t) =β

2π

∑k∈Z

∑~m

∫ ∞0

dλχλ,~m

(γk)etE

(S)R , (2.6.69)

where χλ,~m is the Harish-Chandra character in the principal series of SO (2n+ 1, 1), which hasbeen evaluated [65] to be

χλ,~m (β, φ1, φ2, . . . , φn) =e−iβλχ

SO(2n)~m (φ1, φ2, . . . , φn) + eiβλχ

SO(2n)

~m(φ1, φ2, . . . , φn)

e−nβ∏ni=1 |eβ − eiφi |2

, (2.6.70)

for the group elementγ = eiβQ12eiφ1Q23 . . . eiφnQ2n+1,2n+2 , (2.6.71)

where ~m is the conjugated representation, with the highest weight (m2, . . . ,−mn+1), and χSO(2n)~m

is the character in the representation ~m of SO (2n). The sum over ~m that appears in (2.6.69) isthe sum over permissible values of m as determined by the branching rules (2.5.60). We also recallthat ‘1’ is the time-like direction.

For the thermal quotient, we have β 6= 0 and φi = 0 ∀ i. The character of SO (2n) that appears inthe character formula (2.6.70) above is then just the dimension of the corresponding representation.

26

Using the fact that the dimensions of this representation is equal to that of its conjugate, we havefor the character

χλ,~m (β, φ1, φ2, . . . , φn) =cos (βλ)

22n−1 sinh2n β2

d~m. (2.6.72)

We therefore obtain, for the traced heat kernel AdS2n+1 (2.6.69),

K(S) (β, t) =β

22nπ

∑k∈Z

∑~m

d~m

∫ ∞0

dλcos (kβλ)

sinh2n kβ2

e−t(λ2+ζ). (2.6.73)

The integral over λ is a Gaussian integral, which we can evaluate to obtain

K(S) (β, t) =β

22n√πt

∑k∈Z+

∑~m

d~m1

sinh2n kβ2

e−k2β2

4t−tζ , (2.6.74)

where we have dropped the term with k = 0, which diverges. This divergence arises due to theinfinite volume of AdS, over which the coincident heat kernel on the full AdS2n+1 is integrated.It can be reabsorbed into a redefinition of parameters of the gravity theory under study and isindependent of β and is therefore not of interest to us.

This expression further simplifies for the case of the STT tensors. The branching rules determinethat ~m = (s, 0, . . . , 0) and therefore the sum over ~m gets frozen out and we obtain

K(S) (β, t) =β

22n√πt

∑k∈Z+

d~m

sinh2n kβ2

e−k2β2

4t−t(s+n2). (2.6.75)

The reader may compare this expression to the equation (3.9) obtained in [53] for the AdS3 case(where one would have to specialise to τ1 = 0, τ2 = β, and d~m ≡ 1, and further include a factorof 2 that appears because the branching rule for AdS3 leads us to sum over m2 = ±s rather thanm2 = s for s > 0).

2.7 The one-loop Partition Function

As a consequence of the above, we can calculate the one-loop determinant of a spin-S particle onAdS5. To do so, we need the result that∫ ∞

0

dt

t32

e−α2

4t−β2t =

2√π

αe−αβ. (2.7.76)

Then the one-loop determinants can be deduced from the heat kernel by using

− log det(−∆(S) +m2

S

)=

∫ ∞0

dt

tK(S) (β, t) e−m

2St, (2.7.77)

which we can simplify to obtain

− log det(−∆(S) +m2

S

)=∑k∈Z+

∑~m

d~m2

e−nkβ (ekβ − 1)2n

1

ke−kβ√ζ+m2

S . (2.7.78)

This expression further simplifies for the case of STT tensors and one obtains

− log det(−∆(S) +m2

S

)=∑k∈Z+

d~m2

e−nkβ (ekβ − 1)2n

1

ke−kβ√s+n2+m2

S . (2.7.79)

27

2.7.1 The scalar on AdS5

Let us evaluate the above expression for scalars in AdS5, where s = 0. In units where the AdS

radius is set to one,√m2S + 4 = ∆ − 2, where ∆ is the conformal dimension of the scalar. We

therefore have

− log det(−∆(S) +m2

S

)=∑k∈Z+

2

k (1− e−kβ)4 e−kβ∆ (2.7.80)

We can evaluate the sum to find that the one-loop determinant is given by

− log det(−∆(S) +m2

S

)= −2

∞∑n=0

(n+ 1) (n+ 2) (n+ 3)

6log(

1− e−β(∆+n)). (2.7.81)

Now since logZ(S) = −12 log det

(−∆(S) +m2

S

), we have, for the one-loop partition function of a

scalar,

logZ(S) = −∞∑n=0

(n+ 1) (n+ 2) (n+ 3)

6log(

1− e−β(∆+n)). (2.7.82)

This expression matches exactly with that which is obtained with the method of [66] (see also,earlier work by [67, 68]).

2.8 An extension to Even Dimensions

We have so far considered the case of the odd-dimensional hyperboloids. This is mainly because wehave obtained the heat kernel answer by means of an analytic continuation from the sphere whichessentially captures the ‘principal series’ contribution to the heat kernel. For the odd-dimensionalcase there is no other contribution. The case of even-dimensional hyperboloids is however a bitmore subtle. There can in principle be a contribution from the discrete series also. However, as weoutline below, this series does not contribute for a wide class of tensor fields. In particular, thisincludes the case of the STT tensors which have been of special interest to us7. The answer for theheat kernel in such cases is again captured by the usual analytic continuation, as was explicitlyshown in [64].

We will now briefly sketch how the computation of the traced heat kernel would proceedfor the even-dimensional hyperboloids. We begin by observing that the expression (2.4.54) isvalid for cosets of compact Lie groups G and H, which therefore includes the case of the even-dimensional spheres also. That a thermal section on such spheres may be defined is apparentvia the geometric construction outlined in the main text8. The expression for the heat kernel onS2n ' SO (2n+ 1) /SO (2n), n ≥ 2 is then a sum of characters of the ‘thermal’ quotient group Γembedded in SO (2n+ 1).

The hyperboloid AdS2n is the quotient space SO (2n, 1) /SO (2n). The principal series of unitaryirreducible representations of SO (2n, 1) are labelled by an array

R = (iλ,m2,m3, · · · ,mn) , λ ∈ R, m2 ≥ m3 ≥ · · · ≥ mn, (2.8.83)

where the m2, · · · ,mn are non-negative (half-)integers. These contain a representation S ofSO (2n) if [58, 63]

s1 ≥ m2 ≥ s2 ≥ . . . ≥ mn ≥ |sn|. (2.8.84)

The analytic continuation from the unitary irreducible representations of SO (2n+ 1) to the prin-

7These remarks are true for higher (than two)-dimensional hyperboloids. There is an additional discrete seriescontribution in AdS2 even for the STT tensors. See [64] for details.

8As an example, we see that setting φ3 = 0 in (2.4.35) gives us a parametrisation of S4. The construction of thethermal section is then exactly analogous.

28

cipal series of SO (2n, 1) may be deduced as in Section 2.5.2. The answer is

m1 7→ iλ− 2n− 1

2, λ ∈ R+, (2.8.85)

which is precisely the continuation obtained in [64]. There is in addition an additional discreteseries of representations which is not captured by this analytic continuation. However, this containsthe representation S only if sn ≥ 1

2 , see [58] for details. Therefore, this additional series nevercontributes for the STT tensors (for which sn equals zero). The naive analytic continuation istherefore sufficient to give the full heat kernel answer.

The methods outlined in Sections 2.5 and 2.6 may therefore be extended to even dimensionalhyperboloids as well. The expressions for the global characters of SO (2n, 1) are well known [65].We therefore have all the ingredients needed to compute the heat kernel on thermal AdS2n.

For example, in this manner the one-loop partition function for a scalar on AdS4 may be calcu-lated. We find that

logZ(S) =∑k∈Z+

1

k (1− e−kβ)3 e−kβ∆, (2.8.86)

where ∆ is determined in terms of the mass of the scalar via

∆ =

√m2 +

9

4+

3

2. (2.8.87)

This matches, for instance, with the expressions obtained via the Hamiltonian analysis done in[69].

2.9 Conclusions

We have computed the principal ingredients that go into the calculation of one loop effects on odddimensional thermal AdS spacetimes. In the following two chapters, we shall apply these resultsto evaluate the one-loop partition function for higher-spin theories in AdS spacetimes.

29

30

Chapter 3

The Partition Function for Higher-SpinTheories in AdS

3.1 Introduction

Studies of the asymptotic symmetries of higher-spin theories in AdS3 have recently led to remark-able progress in their holography, including the formulation of an explicit duality between Vasilievtheories in AdS3 and WN minimal models [29]. It was shown in [27, 28] that the asymptoticsymmetry algebra of (a class of) higher-spin theories comprises of two copies of a W algebra.This was checked in [70] by computing the one-loop partition function of the theory, to obtaina vacuum character of left- and right-moving W algebras. The results of [70] were in fact a keyingredient in formulating the duality of [29]. The analysis carried out in [70] drew heavily on theheat kernel methods of [53]. Essentially, the message to take away from these computations is thatthe second-quantised partition function contains information about additional symmetries whichact on multiparticle states of the theory.

While a Brown-Henneaux like analysis for Vasiliev theories in dimensions higher than three isprohibitive, the one-loop partition function may be computed readily. The essential ingredientsare again the quadratic action for higher-spin fields, which is well known (see [28] for a recentreview), and the determinants of the Laplacian, which have been computed in [71].

In this chapter, we shall carry out precisely such a computation for a Vasiliev theory in general-dimensional AdS spacetimes. In particular, we shall evaluate this partition function on a thermalquotient of AdS. As mentioned above, for the one-loop partition function of a theory the detailsof the interactions are not important. The only information that enters is the quadratic actionand the spectrum. We shall choose the spectrum of all spins s = 0, 1, 2, . . .∞ appearing once 1.We shall obtain a form for the partition function which is suggestive of a vacuum character of asymmetry group. We also express the partition function in a form (3.2.17) which is closely relatedto the d-dimensional MacMahon function.

A brief overview of this chapter is as follows. In section 3.2 we compute the partition function ofa massless spin-s field in AdSD by the heat kernel method.2 We show that the degrees of freedomthat contribute to the one-loop partition function are encoded in symmetric, transverse, traceless(STT) tensors of rank s, and s− 1. These determinants have been evaluated in [71] We obtain ananswer consistent with the the partition function of the spin-s conserved current in CFTD−1. Aswe outline later (see the discussion just above section 3.2.1), while this analysis goes through forgeneral dimensional AdS spacetimes, though the expressions for AdS5 are perhaps the nicest, andwe concentrate mostly on this case here.

For evaluating these determinants to arrive at concrete expressions for the partition function,we begin with the relatively simpler case of thermal AdS5, i.e. turning off all other chemical

1As is well known, there is a also minimal Vasiliev theory with only even spins. We consider the theory containingall spins in its spectrum. On the boundary CFT, this corresponds to considering conserved currents built out ofcomplex scalars. Also see [72, 73] for representations of the higher-spin algebra in AdS5.

2We evaluate these determinants explicitly for odd-dimensional AdS spacetimes. For a very general class of tensors,which includes STT tensors, the analysis for even dimensions greater than two is entirely analogous and we donot repeat it here. See [71] and references therein.

31

potentials, in section 3.2.1. We observe simplifications due to which the partition function of thetheory arranges itself into a form reminisctive of the MacMahon function, and also write downa nested, vacuum character like form for this partition function. Expressions for the partitionfunction of the higher-spin theories in AdS4 and AdS7 are available in Appendix C.

In section 3.3 we generalise to the case of non-zero chemical potentials for the SO(4) Cartans ofthe AdS5 isometry group. We find that the essential simplifications in section 3.2.1 remain, dueto which it is still possible to write the answer in terms of a (shifted) two-dimensional MacMahonfunction. We also obtain a nested form for the answer, suggestive of a vacuum character interpre-tation. We then discuss our results in section 3.4. Some calculational details are available in theAppendices.

3.2 The Partition Function for a Higher-Spin Field in AdS

We begin with computing the partition function of a massless spin-s field in an arbitrary dimen-sional (Euclidean) AdS spacetime. Our starting point will be the functional integral

Z(s) =1

Vol(gauge group)

∫ [Dφ(s)

]e−S[φ(s)], (3.2.1)

which we shall compute in the one-loop approximation. Hence, only the spectrum of the theoryand the quadratic terms in the action are of relevance to us. The quadratic action for higher-spinfields has already been worked out in an arbitrary dimensional AdS spacetime and is mentionedin (1.3.26).

We now evaluate the functional integral (4.3.5) with the action (1.3.24) using the method adoptedin [70, 74, 75] which we now summarise. Since this action has a gauge invariance associated withit, we need to gauge fix. Typically this involves the introduction of Faddeev-Popov ghosts. In thiscase however, there is a natural choice of integration variables for the functional integral. We usethe decomposition

φµ1...µs = φTTµ1...µs + g(µ1µ2 φµ3...µs) +∇(µ1ξµ2...µs), (3.2.2)

where φ and ξ are respectively symmetric traceless rank s− 2 and s− 1 tensors. We note that thelast term is just the gauge transformation of the field φ under which the action 1.3.24 is invariant.This is the choice of integration variables employed in [74] for the spin-2 field, and subsequentlygeneralised in [70] to higher-spin fields in AdS3. The measure for the functional integral changesunder this change of variables.

[Dφ(s)] = Z(s)gh [DφTT(s) ][Dφ(s−2)]

[Dξ(s−1)

], (3.2.3)

where Z(s)gh is the ghost determinant that arises from making the change of variables. Now, using

the gauge invariance of the action, we have

S[φ(s)

]= S

[φTT(s) + gφ(s−2)

]. (3.2.4)

One may further show that

S[φTT + gφ

]= S

[φTT

]+ S

[gφ], (3.2.5)

where

S[φTT

]= φµ1...µsTT

[− s2 + (D − 6) s− 2 (D − 3)

`2

]φTTµ1...µs . (3.2.6)

32

Therefore, the contribution to the one-loop partition function is given by

Z(s) = Z(s)gh

[det

(−+

s2 + (D − 6) s− 2 (D − 3)

`2

)(s)

]− 12 ∫ [

Dφ(s−2)

]e−S[φ(s−2)], (3.2.7)

where the subscript (s) on the second term reminds us that the determinant should be evaluatedover rank s STT tensors. This expression should be compared to Equation (2.8) of [70]. Also, theghost determinant may be evaluated using the identity

1 =

∫ [Dφ(s)

]e−〈φ(s),φ(s)〉 = Zgh

∫ [Dφ(s)

] [Dφ(s−2)

] [Dξ(s−1)

]e−〈φ(s),φ(s)〉, (3.2.8)

as in [70, 74]. The evaluation is entirely analogous to the procedure adopted in [70], and we merelymention the final result. The partition function Z(s) is determined to be a ratio of functionaldeterminants evaluated over STT tensor fields. In particular,

Z(s) =

[det(−− (s−1)(3−D−s)

`2

)(s−1)

] 12

[det(−+ s2+(D−6)s−2(D−3)

`2

)(s)

] 12

. (3.2.9)

The numerator is a determinant evaluated over rank s− 1 STT tensor fields. These determinantswere evaluated explicitly in [71], and reviewed in the previous chapter, for Laplacians over quotientsof AdS. In particular, for the thermal quotient of AdS2n+1, we find that

logZ(s) =∞∑m=1

e−mβ(s+2n−2)

(1− e−mβ)2n

(ds − ds−1e

−mβ), (3.2.10)

where we adopt the notation ds for the dimension of the (s, 0, . . . , 0) representation of SO (2n). Wewould like to exponentiate this expression to determine the partition function Z(s). The followingidentity is useful for this purpose.

∞∑m=1

q−m∆

m (1− qm)2n =∞∑m=1

(2n+m− 2

2n− 1

)log

1

1− q(∆+m−1), (3.2.11)

where we have defined q = e−β. Using (3.2.11), we find that

Z(s) =∞∏m=1

((1− q(s+m+2n−2)

)ds−1(1− q(s+m+2n−3)

)ds)(m+2n−2

2n−1 )

. (3.2.12)

This is the contribution of a single spin-s field to the partition function of the theory in an odddimensional AdS spacetime. The case of even dimensions is similar, see Appendix C. Now, it willbe apparent to the reader that many of the subsequent steps that we shall now undertake willfollow in arbitrary dimensions as well. However, the resulting expressions are perhaps the nicestfor the case of AdS5 and we shall focus on this case from now onwards. We refer the reader toAppendix C for the expressions for the blind partition function in AdS4 and AdS7.

3.2.1 The Blind Partition Function in AdS5

While we shall finally work with nonzero chemical potentials for the SO(4) Cartans of AdS5 aswell, it is useful, for intuition, to start with the case where they are zero, i.e. thermal AdS5. This

33

will also yield some curious forms for the partition function. We have, on setting n = 2 in (3.2.12),

Z(s) =∞∏m=1

(1− q(s+m+2)

)s2(1− q(s+m+1)

)(s+1)2

(m+23 )

. (3.2.13)

This is the contribution of a single spin-s field to the partition function of the theory. To determinethe full partition function, we need to specify the spectrum of the theory. We choose the spectrums = 0, 1, 2, . . . ,∞, with each spin appearing only once. The full partition function of the theory isthen

Z =∞∏s=0

Z(s). (3.2.14)

Note that there are cancellations between the numerator of Z(s) and the denominator of Z(s+1),for s ≥ 1 because they are both of the form

(1− q(s+m+2)

)α. We finally obtain

Z =

∞∏s=0

∞∏m=1

1

(1− qs+m+2)4(s+1)(m+23 )·∞∏m=1

1

(1− qm+1)(m+2

3 ). (3.2.15)

We can assemble the first product into a single product over k ≡ s+ n. This is given by

∞∏s=0

∞∏m=1

1

(1− qs+m+2)4(s+1)(m+23 )

=∞∏k=1

1

(1− qk+2)4(k+4

5 ). (3.2.16)

We then obtain

Z =∞∏k=1

1

(1− qk+2)4(k+4

5 )·∞∏m=1

1

(1− qm+1)(m+2

3 ). (3.2.17)

This form of the answer is closely related to the d-dimensional MacMahon function (see [76] for arecent review)

Md (q) =∞∏n=1

1

(1− qn)(n+d−2d−1 )

. (3.2.18)

We remind the reader that in [70] a form for the partition function of a higher-spin theory in AdS3

was obtained in terms of the (two-dimensional) MacMahon function and bore an interpretation asthe vacuum character of WN , the asymptotic symmetry algebra of the theory [27, 28]. We alsoobserve the existence of a ‘nested product’ form for this answer.

Z =∞∏s1=3

∞∏s2=s1

∞∏s3=s2

∞∏s4=s3

∞∏s5=s4

∞∏n=s5

1

(1− qn)4 . (3.2.19)

The reader may verify that this expression does indeed reduce to (3.2.17). We also remind thereader that an analogous expression for AdS3 was interpreted in [70] as the vacuum character ofthe W∞ algebra. It would be very interesting to see if our five-dimensional answer also admits asimilar interpretation.

3.2.2 Additional Chemical Potentials in AdS5

From now onwards we turn on chemical potentials (β, α1, α2) for AdS5, where β is the inversetemperature and α1, α2 are chemical potentials for the SO (4) Cartans of the AdS5 isometry groupSO(4, 2). To explicitly evaluate (3.2.9), we will need the character given in equation 5.3 of [71],

34

where we specialise to n = 2, i.e. AdS5, and focus on the case of STT tensors.

χ(λ,s) (β, α1, α2) =e−iβλχ

SO(4)(s,0) (φ1, φ2) + eiβλχ

SO(4)(s,0) (φ1, φ2)

e−2β|eβ − eiφ1 |2|eβ − eiφ2 |2. (3.2.20)

We can evaluate these characters to obtain

χ(λ,s) (β, α1, α2) =2 cos (βλ)

e−2β|eβ − eiφ1 |2|eβ − eiφ2 |2sin ((s+ 1)α1)

sin (α1)

sin ((s+ 1)α2)

sin (α2), (3.2.21)

where the φs and αs are related by

φ1 = α1 + α2, φ2 = α1 − α2. (3.2.22)

The partition function of a massless spin-s particle may then be computed to obtain

logZ(s) =∞∑m=1

1

m

e−mβ(s+2)

|1− e−m(β−iφ1)|2|1− e−m(β−iφ2)|2[χSO(4)(s,0) − χ

SO(4)(s−1,0)e

−mβ], (3.2.23)

where the SO(4) characters are evaluated over the angles (mφ1,mφ2). This is precisely the answerthat would be obtained via a Hamiltonian computation, as carried out in [69] using the results of[77, 78].

3.3 The Refined Partition Function in AdS5

Having acquired some intuition for the kind of simplifications we may expect by looking at theblind partition function, we shall now now turn to the case of non-zero chemical potentials α1 andα2 and exponentiate the expression (3.2.23). This will enable us to write down another nestedform for the partition function. It would be again useful to introduce notation

e−β = q, eiα1 = p, eiα2 = r. (3.3.24)

We finally obtain

Z(s) =∞∏

mi=0

∏s−1k,l=0

(1− qm1+m2+m3+m4+s+3pm1+m3+s−1rm1+m4+s−1pm2+m4+2krm2+m3+2l

)∏sk,l=0 (1− qm1+m2+m3+m4+s+2pm1+m3+srm1+m4+spm2+m4+2krm2+m3+2l)

,

(3.3.25)where the product over mi collectively denotes a product over m1, . . . ,m4. It turns out that evenfor the refined case, the ratio of the numerator of Z(s) and the denominator of Z(s+1) is againsimple. This leads to the following form for the one-loop partition function of the theory.

Z =

∞∏s=0

∏mi=0

1∏(1− qm1+m2+m3+m4+s+3pm1+m3+s−1rm1+m4+s−1pm2+m4+2krm2+m3+2l)

, (3.3.26)

where the hatted product in the denominator runs over the following values of k, l,

k = 0, l = 0; k = 0, l = s+ 1; k = s+ 1, l = 0 k = s+ 1, l = s+ 1

k = 0, l = 1 . . . s; l = 0, k = 1 . . . s; k = s+ 1, l = 1 . . . s; l = s+ 1, k = 1 . . . s,

and we have used the fact that (pp)2(rr)2 = 1. It is useful to define the following combinations

q1 = pr, q2 = pr, q3 = pr, q4 = pr. (3.3.27)

35

The expression (3.3.26) may be simplified using the procedure outlined in Appendix D. We finallyfind that the partition function may be written in terms of the product of four nested productexpressions.

Z ≡ Z(0) · Z(1) · Z(2) · Z(3) · Z(4), (3.3.28)

where Z(0) is the partition function of the scalar, which may be computed to obtain

Z(0) =∏

m1,2,3,4

1

(1− q2 (qq1)m1 (qq2)m2 (qq3)m3 (qq4)m4), (3.3.29)

and

Z(1) =∞∏k=0

∞∏m2,3,4

∞∏s=1

∞∏n=s

1(1− (qq1)n qk+m2+m3+m4+2qk1q

m2+k2 qm3+k

3 qm44

) . (3.3.30)

This has been explicitly evaluated in appendix D. The other Z(i)s have similar expressions whichwe now enumerate.

Z(2) =

∞∏k=0

∞∏m1,2,4

∞∏s=1

∞∏n=s

1(1− (qq3)n qk+m1+m2+m4+2qm1

1 qm2+k2 qk3q

m4+k4

) , (3.3.31)

Z(3) =∞∏k=0

∞∏m1,2,3

∞∏s=1

∞∏n=s

1(1− (qq4)n qk+m1+m2+m4+2qm1+k

1 qm22 qm3+k

3 qk4

) , (3.3.32)

Z(4) =

∞∏k=0

∞∏m1,3,4

∞∏s=1

∞∏n=s

1(1− (qq2)n qk+m1+m3+m4+2qm1+k

1 qk2qm33 qm4+k

4

) . (3.3.33)

3.4 Discussion

In this chapter, we applied the heat kernel results of [71] to compute the partition function of ahigher-spin theory in AdS5. To get a better feeling for the possible content of the answer it isuseful to recollect elements of the corresponding story for pure gravity and higher-spin gravity inAdS3. This discussion is qualitative, and we refer the reader to the original papers for more details.As we remarked previously, the asymptotic symmetry algebra of pure gravity on AdS3 comprisesof two copies of the Virasoro algebra [7] and for higher-spin gravity comprises of two copies of theW algebra [27, 28], while the gauge symmetry is SL(2, R) × SL(2, R) and SL(N,R) × SL(N,R)or hs(1, 1) respectively. One manifestation of the asymptotic symmetry algebra is in the one-looppartition function of the theories expanded about their AdS vacua. For the higher-spin theorywith spectrum of spins s = 2, . . . ,∞, the partition function is [70]

Z =∞∏s=2

∞∏n=s

1

|1− qn|2= χ0 (W∞)× χ0 (W∞) . (3.4.34)

This is (two copies of) the character evaluated over the Verma module built out of the AdS3

vacuum as the lowest weight state and the raising operatorsW(s)−m, s = 2, . . . ,∞, acting on it. This

is the sense in which the second-quantised partition function is supposed to encode informationabout multiparticle symmetries of the theory. The corresponding expressions for pure gravity[53, 60, 79] are perhaps more familiar to the reader. The partition function is then a character ofthe Virasoro algebra, again with the AdS3 vacuum as the lowest weight state and the generatorsL−m acting as raising operators, for m ≥ 2. These generators, when acting on the AdS3 vacuumgenerate large gauge transformations to create multiparticle states—boundary gravitons—over theAdS3 vacuum.

36

We obtained a form for the answer that is suggestive of a vacuum character of a larger symmetrygroup than the higher-spin symmetry. In particular, following the reasoning suggested above, itleads us to speculate that there are additional generators in the symmetry algebra that act onmultiparticle states built out of twist-two operators, which sit in the vacuum representation ofthe symmetry algebra. It would be very interesting to verify if this is indeed the case. A naturalcandidate symmetry to relate these results to is the Yangian symmetry of N = 4 SYM. However,we do not know of an apparent connection, especially since the Yangian algebra does not closeover the set of gauge-invariant operators.3 Alternatively, this might be related to the additionalsymmetries of free Yang-Mills noted in [80]. There is additionally the intriguing possibility thatthese might be connected to multiparticle symmetries already noted in the literature for higher-spintheories [81–83].4

Additionally enhanced symmetries for higher-spin theories have also been observed in [84–86] .5

It is also possible that our results are related to these symmetries.

3We thank N. Beisert for this remark.4We thank M. Vasiliev for this observation.5We thank D. Sorokin for bringing this to our attention.

37

38

Chapter 4

The Partition Function for TopologicallyMassive Higher-Spin Theory

4.1 Three-Dimensional Gravity and its Topologically Massivedeformation

Possibly the most viable testing ground for theories of quantum gravity has been three dimensionalgravity in Anti de-Sitter space. Pure Anti de-Sitter gravity in three dimensions, as is well known,has no local propagating degrees of freedom. At the same time, it has non-trivial solutions,including Black Holes [89]. This opens up the possibility of using three-dimensional gravity as atoy model for studying various interesting phenomena such as the black hole information paradox.At the same time, three-dimensional gravity is a particularly promising arena to probe otherinteresting questions about quantum gravity, such as the roles of time, topology and topologychange in qauntum gravity. We refer the reader to [90] for a review of these aspects of threedimensional quantum gravity.

Motivated by the AdS/CFT conjecture, and the above considerations, there have been attemptsto find the dual of the 3d AdS theory in terms of a conformal field theory. The field theory dualto the bulk Einstein theory in AdS3 was initially conjectured to be an extremal CFT [91]. But apartition function computation by taking into account contributions from classical geometries andalso including quantum corrections [79] showed that the expected holomorphic factorisation doesnot hold and there were other studies [92] which indicated that this conjecture was incorrect.

Motivated by [91], the authors of [93] looked at Topologically Massive Gravity (TMG) in AdS3,which consists of Einstein gravity, modified by the addition of a gravitational Chern-Simons term[87, 88]

S3 = SEH + SCS (4.1.1)

where SEH =

∫d3x√−g(R− 2Λ), (4.1.2)

and SCS =1

2µ

∫d3xεµνρ

(Γσµλ∂νΓλρσ +

2

3ΓσµλΓλνθΓ

θρσ

). (4.1.3)

This theory already has the appealing feature that in contrast to usual Einstein gravity in AdS3,it has local propagating gravitational degrees of freedom, thus making it more like the higher-dimensional theories.1 However, a closer look reveals some disquieting features. In particular, it

1The theory is of course, parity violating, in contrast to Einstein gravity in higher dimensions.

39

turns out that at a generic value of µ, the theory has negative energy gravitons. 2 However, itturns out that at the ‘chiral’ point µ` = 1 the negative energy gravitons (which are purely left-moving) have zero energy, and are thus pure gauge. Also, at this point, the corresponding centralcharge c± = 3l

2G(1∓ 1µ`) [94, 95] vanishes. This therefore gives rise to a holomorphically factorised

(trivially so, since its purely right-moving) partition function. It was therefore conjectured in [93]that the boundary theory dual to TMG in AdS3 is a right-moving chiral CFT.

This conjecture recieved intense scrutiny (see, for example, the works [96]), and in [97, 98] it wasshown that TMG at the chiral point was more generally dual to a Logarithmic Conformal FieldTheory (LCFT) and that there were additional normalisable solutions to TMG which carriednegative energy at the chiral point, thus invalidating the earlier conjecture. A more completeanalysis in terms of holographic renormalisation techniques [99] was carried out and the resultssupported the latter claim of the duality to LCFT. One of the more robust checks of this conjecturewas the recent computation of the one-loop partition function [74]. It was conclusively shown thatthere is no holomorphic factorisation of the one-loop partition function at the chiral point. Thestructure of the gravity calculation also matched with expectations from LCFT. The authors [74]found an exact matching with a part of the answer, viz. the single-particle excitations and offeredsubstantial numerical evidence of the matching of the full partition functions.

4.2 Higher-Spin Theories in AdS3

Much as in the case of gravity, three-dimensional higher-spin theories are a useful playground forstudying dynamics of higher-spin fields in AdS spacetimes. In contrast to the higher-dimensionalcases, in three dimensions there can be a truncation to fields with spin less than and equal toN for any N .3 It has been argued in [27, 28] on the basis of a Brown-Henneaux analysis thatclassically, these theories have an extended classical WN asymptotic symmetry algebra (see alsothe recent work [100]). A one-loop computation in [70], using the techniques developed in [53],showed that this symmetry is indeed perturbatively realised at the quantum level as well. Thiswas an important ingredient in formulating the duality [29] between higher-spin theories and WN

minimal models in the large-N limit. There was also a subsequent work [101], in which the authorsprovided a bound on the amount of higher spin gauge symmetry present. This was regarded as agravitational exclusion principle, where quantum gravitational effects place an upper bound on thenumber of light states in the theory. Different tests of this duality have been performed successfullysubsequent to its proposal, see [102].

The higher spin theories described above, like Einstein AdS gravity in three dimensions, do nothave any propagating degrees of freedom in the ‘gravitational’ sector.4 It is natural thus to ask ifone can generalise Topologically Massive Gravity to theories of higher spin. A construction of thistheory, called Topologically Massive Higher Spin Gravity (TMHSG) was initated in [103, 104]. Asa first step to this end, the quadratic action for a spin-3 field was studied in the linearised approxi-mation about AdS. The results of this analysis were strongly suggestive of the interpretation thatthe spin-3 theory at the so-called chiral point is dual to a logarithmic CFT. Specifically, the spaceof solutions developed an extra logarithmic branch at the chiral point.

In this chapter, we shall describe a test of this conjecture at the (leading) quantum level. Inparticular, we shall compute the one-loop partition function of the spin-3 theory on a thermalquotient of AdS3 and show that it does not factorize holomorphically at the chiral point. Ouranalysis is along the lines of [74], and our results may be viewed as a higher-spin generalisation

2We adopt sign conventions such that black holes have positive energy.3It is also notable that these simpler theories with a finite number of spins are consistent truncations of the more

non-trivial Vasiliev theories in AdS3.4The Vasiliev theories to which the WN minimal models are believed to be dual contain propagating scalars. In

contrast to higher-dimensions, however, the scalars fall into separate representations of the higher-spin algebrathan the spins two and higher.

40

of theirs. We shall also compute the one-loop partition function for a spin-N generalisation ofthe action proposed in [103] and show that this property continues to hold. We interpret theseresults as an indication that the dual CFT at the chiral point is not chiral, and that the resultsare consistent with the expectation of a high spin extension of a dual logarithmic CFT.

In particular, we find the contribution to the one loop partition function at the chiral point,from the spin-3 fields to be

Z(3)TMHSG =

∞∏n=3

1

|1− qn|2∞∏m=3

∞∏m=0

1

(1− qmqm)

∞∏k=4

∞∏k=3

1(1− qkqk

) , (4.2.4)

where q ≡ eiτ , and τ is the modular parameter on the boundary torus of thermal AdS3. Thefirst term is the holomorphic contribution determined from a study of the massless theory in [70],the other terms are non-holomorphic, and new. The middle term is the contribution from thetransverse traceless spin-3 determinant, while the last term is the transverse spin-1 contributioncoming from the trace of the spin-3 field.

The case of general spins yields analogous results. The relevant excitations are transverse trace-less spin-s, spin-s − 1 (coming from the ghost determinant), transverse traceless spin-s − 2 ones(which is the trace of the spin-s field), transverse traceless spin-s−3 (coming from the longitudinalcomponent of trace), transverse traceless spin-s−4 (coming from the longitudinal component of thelongitudinal component of the trace) and so on upto transverse traceless spin-1, of which the spin-s, s− 2, s− 3, · · · , 1 contribute non-holomorphically to the one-loop partition function. There areno other relevant excitations coming from the spin-s analysis because of the double-tracelessnesscondition reviewed, for example, in [28]. In section 4.6, we conclude with a brief interpretation ofour results. We will also do a classical analysis for arbitrary spins in appendix E and show thatthe contributions to the partition function also appear in the classical spectrum.

4.3 The basic set up for s = 3

In this section we will compute the one loop partition function for spin 3 TMHSG, and in theprocess build up a mechanism to generalise our calculations to arbitrary spin in the subsequentsection. Following the method adopted in [70], we shall compute the one-loop partition functionin the Euclideanised version of theory via the path integral

Z(s) =1

Vol(gauge group)

∫ [Dφ(s)

]e−S[φ(s)]. (4.3.5)

In the one-loop approximation, only the quadratic part of the action S[φ(s)

]is relevant. This has

been worked out for TMHSG, for the case of s = 3, in [103]. In the Euclidean signature it takesthe form 5

S =1

2

∫d3x√gφMNP

[FMNP −

1

2F(MgNP )

], (4.3.6)

where

FMNP = D(M)FMNP ≡ FMNP +i

6µεQR(M∇QFRNP ), (4.3.7)

and

FMNP = ∆φMNP −∇(M |∇Qφ|NP )Q +1

2∇(M∇NφP ) −

2

`2g(MNφP ). (4.3.8)

As always, the brackets “( )” denote the sum of the minimum number of terms necessary toachieve complete symmetrisation in the enclosed indices without any normalisation factor. Let

5This is the most general parity violating, three derivative action for the free spin-3 field. It can also be obtainedfrom linearising the parity violating, non-linear theory of [104]

41

us also define the operation of D(M) on the trace of the spin-3 field by taking the trace of theexpression in (4.3.7)

D(M)φM = φM +i

6µεQRM∇QφR (4.3.9)

We now decompose the fluctuations φMNP into transverse traceless φ(TT ), trace φM and longitu-dinal parts ∇(MξNP ) as6

φMNP = φ(TT )MNP + φ(MgNP ) +∇(MξNP ). (4.3.10)

Following [70], we use gauge invariance, and orthogonality of the first two terms in (4.3.10) todecompose the action (4.3.6) as

S[φMNP ] = S[φ(TT )MNP ] + S[φM ], (4.3.11)

where

S[φ(TT )MNP ] = −1

2

∫d3x√gφ(TT )MNP

[−D(M)∆

]φ

(TT )MNP , (4.3.12)

and

S[φM ] =9

4

∫d3x√g

[8φMD(M)

(−∆ +

7

`2

)φM − φMD(M)∇M∇QφQ

]. (4.3.13)

Now one can further decompose φM into its transverse and longitudinal parts as

φM = φ(T )M +∇Mχ. (4.3.14)

The action for φM then becomes

S[φM ] =9

4

∫d3x√g

[8φ(T )MD(M)

(−∆ +

7

`2

)φ

(T )M + 9χ

(−∆ +

8

`2

)(−∆)χ

]. (4.3.15)

We see that the χ part of the action is same as that obtained for the massless theory in [70] andwill subsequently cancel with the relevant term coming from the ghost determinant, which arisesfrom making the change of variables (4.3.10) in the path integral (4.3.5). The ghost determinant–being independent of the structure of the action– is essentially be the same as that obtained [70],see their expression (2.14). This turns out to imply that although that the trace contribution doesnot cancel with the ghost determinant, the contribution coming from its longitudinal part does.This is in accordance with the observation in [103] that the trace in TMHSG is not pure gaugeunlike the massless theory. It is however still true even in the topologically massive theory that∇MφM is pure gauge and hence the longitudinal contribution from the trace φM does cancel withthe ghost determinant.

The spin-3 contribution to the one loop partition function for TMHSG is given by

Z(3)TMGHS = Z

(3)gh × (det[(−D(M)∆)]TT(3) )−

12 × (det[D(M)(−∆ +

7

`2)]T(1))

− 12

× (det[−∆(−∆ +8

`2)](0))

− 12 , (4.3.16)

where Z(3)gh is the ghost determinant arising as a Jacobian factor corresponding to the change of

variables (4.3.10). The ghost determinant has been obtained in [70], see their expression (3.9), andis given by

Z(3)gh = [det(−∆ +

6

`2)TT(2) × det(−∆ +

7

`2)T(1) × det(−∆ +

8

`2)(0)]

12 . (4.3.17)

6Note that this decomposition is not orthogonal, in the sense that the trace part contains longitudinal terms, andthe longitudinal part is not traceless. See [70].

42

Therefore, the spin-3 contribution to the one loop partition function is given by

Z(3)TMGHS = [det[−D(M)∆]TT(3) ]−

12 [det[D(M)]

T(1)]− 1

2 [det[−∆ +6

`2)]TT(2) ]

12

≡ Z(3)masslessZ

(3)M , (4.3.18)

where Z(3)massless is the massless spin-3 partition function obtained in [70], which is

Z(3)massless =

∞∏s=3

1

|1− qn|2. (4.3.19)

It remains to determine Z(3)M . In order to do so, we shall follow the method of [74], and first

calculate the absolute value |Z(3)M |. Following [74], we define

|Z(3)M | =

[det(D(M)D(M))

TT(3)

]− 14 ×

[det(D(M)D(M))

T(1)

]− 14. (4.3.20)

The subscript (3) and (1) signifies that the operators are acting on transverse traceless spin-3 andtransverse spin-1 respectively. Using (4.3.7) and (4.3.9), we can show that

D(M)D(M)φTTMNP = − 1

4µ2(∆ +

4

`2− 4µ2)φTTMNP

D(M)D(M)φTP = − 1

36µ2[−∆ + (36µ2 − 2

`2)]φTP . (4.3.21)

Therefore,

|Z(3)M | =

[det

(−∆ + 4(µ2 − 1

`2)

)(3)

]− 14

×

[det

(−∆ + (36µ2 − 2

`2)

)(1)

]− 14

. (4.3.22)

4.4 One-loop determinants for spin-3

We shall evaluate the one-loop determinants utilising the machinery developed in [53], accordingto which the relevant determinant takes the following form

− log det(−∆ +m2s

`2)TT(s) =

∫ ∞0

dt

tK(s)(τ, τ ; t)e

−m2(s)t, (4.4.23)

where K(s) is the spin-s heat kernel given by

K(s)(τ, τ ; t) =∞∑m=1

τ2√4πt|sinmτ2 |2

cos(smτ1)e−m2τ22

4t e−(s+1)t. (4.4.24)

We therefore obtain

− log det

(−∆ +

4(µ2l2 − 1)

`2

)TT(3)

=

∞∑m=1

1

m

cos(3mτ1)

|sinmτ2 |2e−2µ`mτ2

=∞∑m=1

2

m

q3m + q3m

(1− qm)(1− qm)(qq)m(µ`−1),

(4.4.25)

43

and similarly,

− log det

[−∆ +

36µ2l2 − 2

`2

]T(1)

=∞∑m=1

2

m

qm + qm

(1− qm)(1− qm)(qq)3µ`m. (4.4.26)

We have used the notations and conventions of [53] throughout. We remind the reader that q ≡ eiτ ,where τ = τ1 + iτ2 is the complex structure modulus on the boundary torus of thermal AdS3. Wecan now put together all the contributions into the expression for the one-loop partition function

log |Z(3)M | = −1

4log det

(−∆ +

4(µ2l2 − 1)

`2

)TT(3)

− 1

4log det

[−∆ +

36µ2l2 − 2

`2

]T(1)

=∞∑m=1

1

2m

q3m + q3m

(1− qm)(1− qm)(qq)m(µ`−1) +

∞∑m=1

1

2m

qm + qm

(1− qm)(1− qm)(qq)3µ`m.

(4.4.27)

Following [74], we infer from this that at the chiral point we have (after rewriting log |Z(3)M | =

12 logZ

(3)M + 1

2 log Z(3)M )

logZ(3)M =

∞∑m=1

1

m

(q3m + (qq)3mqm

)(1− qm) (1− qm)

. (4.4.28)

Let us now note that

∞∑n=1

1

n

q3n

(1− qn) (1− qn)= −

∞∑m=3

∞∑m=0

log(1− qmqm

)∞∑n=1

1

n

(qq)3nqn

(1− qn) (1− qn)= −

∞∑m=4

∞∑m=3

log(1− qmqm

), (4.4.29)

and hence the spin-3 contribution to the full partition function at the ciral point, factorizes as(after including the massless contribution from [70])

Z(3)TMHSG =

∞∏n=3

1

|1− qn|2∞∏m=3

∞∏m=0

1

(1− qmqm)

∞∏k=4

∞∏k=3

1(1− qkqk

) . (4.4.30)

This is the expression (4.2.4) that we had mentioned in the introduction. As noted there, thepartition function does not factorize holomorphically. A chiral CFT would not give rise to sucha partition function. We therefore take this as a one-loop clue that the dual CFT to TMHSG isindeed not chiral, but logarithmic. We remind the reader that an analogous result was found in[74] for the case of topologically massive gravity, from which the authors were able to concludethat the dual CFT was indeed logarithmic.

4.5 Generalisation to arbitrary spin

In this section, we will consider a natural generalisation of our spin-3 action (4.3.6) to an arbitraryspin-s field. We will perform the analysis of Sections 4.3 and 4.4 and show that holomorphicfactorisation does not occur. We take the action7

S =1

2

∫d3x√gφM1M2···Ms

[FM1M2···Ms −

1

2FPP (M1···Ms−2

gMs−1Ms)

], (4.5.31)

7See [105] for a derivation

44

where

FM1···Ms = D(M)FM1M2···Ms ≡ FM1···Ms +i

s(s− 1)µεQR(M1

∇QFRM2···Ms), (4.5.32)

and

FM1M2M3.....Ms ≡ ∆φM1M2.....Ms −∇(M1|∇Qφ|M2.....Ms)Q +

1

2∇(M1

∇M2φP

M3M4....Ms)P

− 1

`2

[s2 − 3s

]φM1M2....Ms + 2g(M1M2

φ PM3M4...Ms)P

.

(4.5.33)

Now we would like to consider the possibility that the higher-spin theory with spins upto N isalso dual to a high spin extension of LCFT. One would then expect that along the lines of whatwe have seen for spin-3, the one-loop partition function again would not factorize holomorphically.We will now do a one-loop analysis using the action (4.5.31) with this goal in mind. An essentialingredient for this analysis – as in the case of spin 3 – is the action of D(M ) on symmetric transversetraceless (STT) tensors of rank s and below. Using the definition of D(M) in (4.5.32), one canshow that

D(M)D(M)φ(TT )M1···Ms

=1

(s− 1)2µ2

[−∆ +

m2s

`2

]φ

(TT ),M1···Ms

D(M)D(M)φ(s−2,TT )M3···Ms

=(s− 2)2

(s− 1)2s2µ2

[−∆ +

m2s−2

`2

]φ

(s−2,TT )M1···Ms

,

D(M)D(M)χ(s−m,TT )Ms−m+1···Ms

=(s−m)2

(s− 1)2s2µ2

[−∆ +

m2s−m`2

]χ

(s−m,TT )Ms−m+1···Ms

, (4.5.34)

where

m2s = µ2`2(s− 1)2 − (s+ 1)

m2s−m = µ2`2

s2(s− 1)2

(s−m)2 − (s−m+ 1), s > m ≥ 2. (4.5.35)

Here φ(s−2) and χ(s−n) are STT tensors of rank s− 2 and s− n respectively, where n ≥ 3.

Following the same steps as for spin-3 in the previous section, the one loop partition functioncan be factorized as

Z(s)TMHSG = Z

(s)(massless)Z

(s)M , (4.5.36)

The contributions to Z(s)M come from the determinants of D(M) acting on φ(TT ), φ(s−2,TT ) and

χ(s−m,TT ) (for s > m ≥ 3). This is apparent from (4.5.34), which suggests that D(M) acts nontrivially on the STT modes of spin-1 and higher. The action of D(M) stops at χ(1) as it actstrivially on scalars, being just the identity map. After a careful analysis, one sees that

log |Z(s)M | = −

1

4log det

(−∆ +

m2s

`2

)(TT )

(s)

− 1

4

s−1∑m=2

log det

(−∆ +

m2s−m`2

)(TT )

(s−m)

. (4.5.37)

45

We find, using the spin-s heat kernel (4.4.24) and the expression (4.4.23) for the determinant,

− log det

(−∆ +

m2s

`2

)(TT )

(s)

=∞∑m=1

2

m(qq)

[m

(s−1)2

(µ`−1)]qms + qms

|1− qm|2,

− log det

(−∆ +

m2s−m`2

)(TT )

(s−m)

=∞∑n=1

2

n(qq)[k(s,m,µ`)n] q

n(s−m) + qn(s−m)

|1− qn|2, (4.5.38)

Where,

k(s,m, µ`) =s(s− 1)µ`− (s−m)(s−m− 1)

2(s−m). (4.5.39)

At the chiral point µ` = 1, it becomes

k(s,m, 1) ≡ k(s,m) =s(s− 1)− (s−m)(s−m− 1)

2(s−m)=m(2s−m− 1)

2(s−m), (4.5.40)

and hence at the chiral point the partition function Z(s)M becomes

logZ(s)M =

∞∑n=1

1

n

(qsn +

∑s−1m=2(qq)k(s,m)nq(s−m)n

)(1− qn) (1− qn)

. (4.5.41)

After using the identities

∞∑n=1

1

n

qsn

(1− qn) (1− qn)= −

∞∑m=s

∞∑m=0

log(1− qmqm

),

∞∑n=1

1

n

(qq)k(s,m)nq(s−m)n

(1− qn) (1− qn)= −

∞∑p=r(s,m)

∞∑p=k(s,m)

log(1− qpqp

)(4.5.42)

where,

r(s,m) = k(s,m) + s−m =s(s− 1) + (s−m)(s−m+ 1)

2(s−m), (4.5.43)

the contribution to the full partition function from spin-s field at the chiral point, becomes

Z(s)TMHSG =

∞∏n=s

1

|1− qn|2∞∏m=s

∞∏m=0

1

(1− qmqm)

s−1∏t=2

∞∏p=r(s,t)

∞∏p=k(s,t)

1

(1− qpqp). (4.5.44)

Hence, the full partition function at the chiral point, in a theory with fields of spin s = 3, · · ·N inaddition to the spin 2 graviton, becomes

ZTMHSG =N∏s=2

[ ∞∏n=s

1

|1− qn|2∞∏m=s

∞∏m=0

1

(1− qmqm)

]×

N∏s=3

s−1∏t=2

∞∏p=r(s,t)

∞∏p=k(s,t)

1

(1− qpqp)

,Where k(s,m) =

s(s− 1)− (s−m+ 1)(s−m− 1)

2(s−m), r(s,m) = k(s,m) + s−m

(4.5.45)

We will see in appendix E that r(s,m) and k(s,m) appear respectively as left and right weights ofclassical left moving primary solution (or massive primary at the chiral point). In particular, wewill also see that, r(s,m, µ`) ≡ k(s,m, µ`) + s−m, and k(s,m, µ`) will be the weights of massiveprimary at a generic point. Thus we also see that for generic spins, our one loop computationsand classical computations are also mutually consistent.

46

Note that the first square bracket contribution starts from s = 2 whereas the second squarebracket contribution starts from s = 3. This is a novel feature of topologically massive higher spintheory and technically this comes from the fact the trace is not a pure gauge. As the spin increases,the factors contributing to the partition function also increases. This, as discussed before, comesfrom the fact that longitudinal component of trace, longitudinal components of the longitudinalcomponents of the trace and so on are not pure gauge and starts contributing till one hits a scalar,which is a pure gauge and the contribution terminates.

One might be worried about the fact that k(s,m) and r(s,m) in (4.5.45) are not integers forgeneric spin, and this might not be consistent with the periodicity in τ . However one might notethat r(s,m)− k(s,m) = s−m, which is an integer and hence the periodicity in τ is not affectedbecause of the following identity

qr(s,m)qk(s,m) = (qq)k(s,m)qs−m. (4.5.46)

4.6 Conclusions

In this chapter, we computed the one loop partition function for topologically massive higher spingravity (TMHSG) for spin-3 and later generalised it to arbitrary spin. We find that the one looppartition function does not factorize holomorphically giving strong evidence that the dual theoryis a high spin extension of LCFT. This was also anticipated in the classical calculation for spin-3TMHSG in [103] as extra logarithmic modes emerged at the chiral point. Additionally there isthe presence of a non trivial spin one contribution (in case of spin 3 TMHSG) in the one looppartition function. This points to the fact that the trace modes cannot be gauged away, both ofwhich results are in accordance with the analysis of classical solutions in [103].

47

48

Part III

Conclusions

Chapter 5

Conclusions and Outlook

In this thesis, we have studied aspects of the holography of higher-spin theories. We were motivatedboth by the central role that higher-spin theories seem to have in unravelling the tensionless limitof string theory as well as the novel holographic dualities that the three-dimensional higher-spintheories admit.

We firstly constructed as a main tool, the heat kernel for the Laplacian on arbitrary-spin fieldson AdS spacetimes in chapter 2. We subsequently used this to gain insight on the multiparticlesymmetries of higher-spin theory in chapters 3 and 4. We provided evidence in chapter 3 thatthe higher-spin theories in higher dimensions, such as AdS5 have additional symmetries that acton the multiparticle states of the theory, much like the enhanced symmetries in AdS3, whichare present even for pure gravity. Finally in chapter 4 we computed the partition function oftopologically massive higher-spin theories. We found that the partition function does not factoriseholomorphically. The overall structure of the answer leads us to expect the theory to be dual to alogarithmic CFT with extended W symmetries.

Finally, we believe that the work in this thesis is only the beginning of a very important line ofexploration. Firstly, it is important to precisely elucidate the nature of the enhanced symmetriesconjectured in chapter 3. This would have important implications for the AdS/CFT duality.Secondly, we would like to completely carry out the program we have initiated for topologicallymassive higher-spin theories to construct a duality between higher-spin theories in AdS3 andlogarithmic CFTs with W symmetry. There are many additional problems to be solved to meetthis overall goal, not the least of which is constructing such CFTs. However, such a duality wouldhave important condensed matter applications and is a significant overall goal of our program.

We would like to return to these questions in the future.

51

52

Part IV

Appendices

Appendix A

Normalising the Heat Kernel on G/H

We determine the normalisation factor a(S)R that arises in the expressions (2.3.16) onwards. This

is fixed by demanding that the integrated traced heat kernel obey∫G/H

dµ (x)K(S) (x, x; t) =∑R

dRetE

(S)R . (A.0.1)

Using the expression (2.3.17) for the heat kernel on G/H, we have∫G/H

dµ (x)K(S) (x, x; t) =∑R

∫G/H

dµ (x) a(S)R U

(R)(σ (x)−1 σ (x)

)a

aetE

(S)R

=∑R

a(S)R VG/HdSe

tE(S)R . (A.0.2)

On comparing (A.0.1) and (A.0.2), we obtain the required relation

a(S)R =

dRVG/HdS

, (A.0.3)

which we use in the main text from (2.3.16) onwards.

55

56

Appendix B

The Plancherel Measure for STT tensors on HN

We show how the normalisation constant aSR gets analytically continued to µSR, the measure forthe λ integration that appears in the AdS heat kernel. Let us consider the expression for d(m,s)

which we obtain from (2.3.29).

dm1,s =

n+1∏j=2

l21 − l2jµ2

1 − µ2j

n+1∏j=3

l22 − l2jµ2

2 − µ2j

. (B.0.1)

The first product in the numerator gets analytically continued via (2.5.57) to

n+1∏j=2

(l21 − l2j

)7→ (−1)n

[λ2 + (s+ n− 1)2

] n−2∏j=0

(λ2 + j2

), (B.0.2)

while the second product evaluates to

n+1∏j=2

(l22 − l2j

)=

(s+ n− 1) (s+ 2n− 3)!

s!. (B.0.3)

The denominator∏n+1j=2

(µ2

1 − µ2j

)∏n+1j=3

(µ2

1 − µ2j

)evaluates to

n+1∏j=2

(µ2

1 − µ2j

) n+1∏j=3

(µ2

1 − µ2j

)= (2n− 2)!

22n−2

√πn!Γ

(n+

1

2

). (B.0.4)

The dimension d(m,s) then gets continued to

(−1)N−1

2

[λ2 +

(s+ N−3

2

)2]∏N−52

j=0

(λ2 + j2

)2N−2√

πΓ(N2

) (N−1

2

)!

(2s+N − 3) (s+N − 4)!

s! (N − 3)!, (B.0.5)

where we have changed variables from n to N = 2n+ 1. We now use the fact that for odd N

VSN =(N + 1)π

N+12

Γ(N+3

2

) =2π

N+12

Γ(N+1

2

) ,= 2πN+1

2(N−1

2

)!

(B.0.6)

and hence the combinationd(m,s)VG/H

gets mapped to

(−1)N−1

2

[λ2 +

(s+ N−3

2

)2]∏N−52

j=0

(λ2 + j2

)2N−1π

N2 Γ(N2

) (2s+N − 3) (s+N − 4)!

s! (N − 3)!. (B.0.7)

57

Using the expressions from [64] quoted in the main text, i.e.

ΩN−1 =2π

N2

Γ(N2

) , cN =2N−2

π, g(s) =

(2s+N − 3)(s+N − 4)!

(N − 3)!s!(B.0.8)

and

µ(λ) =π[λ2 + (s+ N−3

2 )2]∏N−5

2j=0 (λ2 + j2)[

2N−2Γ(N2

)]2 , (B.0.9)

we see that the normalisation constant aSR gets mapped to

µSR =CNg (s)

dS

µ (λ)

ΩN−1, (B.0.10)

where we have omitted the overall sign that appears for some values of N as an artefact of theanalytic continuation, since the measure is always positive definite. The coincident heat kernel istherefore

KS (x, x, t) =

∫dλµSR (λ) dSe

tE(S)R =

CNΩN−1

g (s)

∫dλµ (λ) e−t(λ

2+s+n2). (B.0.11)

Now, the coincident heat kernel may also be written down using 2.7 of [64] (in their notation) as

KS (x, x, t) =∑u

∫ ∞0

dλ hλu∗ · hλu (x) e−t(λ2+s+n2). (B.0.12)

On choosing x to be the origin (which we can do for arbitrary x), and using their expression 2.10,we conclude that

KS (x, x, t) =CN

ΩN−1g (s)

∫dλµ (λ) e−t(λ

2+s+n2), (B.0.13)

which is precisely the expression we have obtained via analytic continuation.

58

Appendix C

Blind Partition functions for AdS4 and AdS7

We shall begin by using (3.2.12) to obtain the blind partition function of the Vasiliev theory inAdS7. Observations parallel to those in Section 3.2.1 lead us to conclude that the expressions forthe partition function are again simple. We need to evaluate the product

∞∏s=1

∞∏m=1

1

(1− qs+m+4)(m+4

5 )(ds+1−ds−1), (C.0.1)

where ds denotes the (s, 0, 0) representation of SO (6). Computing these dimensions (see [71] andreferences therein), we find that the partition function of the Vasiliev theory may be expressed asthe product

Z = Z(0) ·∞∏s=0

∞∏m=1

1

(1− qs+m+4)(m+4

5 )(ds+1−ds−1), (C.0.2)

where Z(0) is the partition function of the scalar, given by

Z(0) =

∞∏m=1

1

(1− qm+3)(m+4

5 ). (C.0.3)

We finally obtain

Z =∞∏m=1

1

(1− qm+3)(m+4

5 )· 1

(1− qm+4)4(m+89 )+(m+6

7 )+(m+56 )

, (C.0.4)

which is again related to the d-dimensional MacMahon function (3.2.18).

We now turn to the case of AdS4. The entire heat kernel analysis of this paper and [71] may beapplied to even dimensional hyperboloids. We merely mention the final result for AdS4. We findthat

Z =1

(1− q) (1− q2)3

∞∏m=2

1

(1− qm)(m+1

2 ) (1− qm+1)3(m+12 )+4(m+2

3 ). (C.0.5)

This is again related to the d-dimensional MacMahon function (3.2.18).

59

60

Appendix D

Computing the Refined Partition Function

We outline some of the main steps involved in reducing (3.3.26) to the nested form (3.3.28). We

begin by expanding out∏

that appears in (3.3.26) into two products A and B. We will evaluate

A =

∞∏mi=0

∞∏s=0

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4 rm2+m3

)∞∏

mi=0

∞∏s=0

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4 rm2+m3+2(s+1)

)∞∏

mi=0

∞∏s=0

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4+2(s+1)rm2+m3

)∞∏

mi=0

∞∏s=0

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4+2(s+1)rm2+m3+2(s+1)

),

(D.0.1)

and

B =∞∏

mi=0

∞∏s=1

s∏k=1

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4 rm2+m3+2k

)∞∏

mi=0

∞∏s=1

s∏k=1

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4+2krm2+m3+2(s+1)

)∞∏

mi=0

∞∏s=1

s∏k=1

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4+2krm2+m3

)∞∏

mi=0

∞∏s=1

s∏k=1

(1− qm1+m2+m3+m4+s+3pm1+m3+s+1rm1+m4+s+1pm2+m4+2(s+1)rm2+m3+2k

).

(D.0.2)

The full partition function of the theory is the inverse of the product of A and B. We will nowsimplify these expressions.

A =∏mi,s

(1− qm1+m2+m3+m4+s+3qm1+s+1

1 qm22 qm3

3 qm44

)(1− qm1+m2+m3+m4+s+3qm1

1 qm22 qm3+s+1

3 qm44

)∏mi,s

(1− qm1+m2+m3+m4+s+3qm1

1 qm22 qm3

3 qm4+s+14

)(1− qm1+m2+m3+m4+s+3qm1

1 qm2+s+12 qm3

3 qm44

),

(D.0.3)

61

where we have used the fact that pp = 1 = rr. In each product, there is a pairing of one of mi ands. We use this to write

A =∞∏

mi=0

(1− qm+s+3qm1+1

1 qm22 qm3

3 qm44

)m1+1∞∏

mi=0

(1− qm+s+3qm1

1 qm22 qm3+1

3 qm44

)m3+1

∞∏mi=0

(1− qm+s+3qm1

1 qm22 qm3

3 qm4+14

)m4+1∞∏

mi=0

(1− qm+s+3qm1

1 qm2+12 qm3

3 qm44

)m2+1,

(D.0.4)

where we have defined m ≡∑

imi. Now each product is of the form

∞∏n=1

(1− αqn)n =

∞∏s=1

∞∏n=s

(1− αqn) , (D.0.5)

where α is independent of n. This gives a (two-dimensional) MacMahon function form for thissector of the partition function, which we can arrange into a nested form. For example, the firstproduct reduces to

∞∏m2,m3,m4=0

( ∞∏s=1

∞∏n=s

(1− (qq1)n qm2+m3+d+2qm2

2 qm33 qm4

4

)). (D.0.6)

There are similar expressions for the other three products, which arise by permuting qi’s above.We will now look at the B term, and obtain its nested product representation. Written in termsof the qis, the first product that enters in B is

∞∏mi=0

∞∏s=1

s∏k=1

(1− (qq1)(m1+s+1) qm2+m3+d+2qm2+k

2 qm3+k3 qm4

4

)=

∞∏mi=0

∞∏k=1

∞∏s=k

(1− (qq1)(m1+s+1) qm2+m3+d+2qm2+k

2 qm3+k3 qm4

4

),

(D.0.7)

where we have changed the products over k and s. This enumerates the same combinations (k, s)as the previous product. Again, redefining a+ s to a, and s− k to s, we get

∞∏mi=0

∞∏k=1

s∏s=0

(1− (qq1)(m1+s+1) qk+m2+m3+m4+2qk1q

m2+k2 qm3+k

3 qm44

)=

∞∏k=1

∞∏mi=0

(1− (qq1)(m1+1) qk+m2+m3+m4+2qk1q

m2+k2 qm3+k

3 qm44

)m1+1,

(D.0.8)

from which we can write the nested form

∞∏k=1

∞∏m2,3,4

∞∏s=1

∞∏n=s

(1− (qq1)n qk+m2+m3+m4+2qk1q

m2+k2 qm3+k

3 qm44

)(D.0.9)

Note that (D.0.6) is just the k = 0 term in this product. We can write the contribution of thesetwo terms together as

Z−1(1) =

∞∏k=0

∞∏m2,3,4

( ∞∏s=1

∞∏n=s

(1− (qq1)n qk+m2+m3+m4+2qk1q

m2+k2 qm3+k

3 qm44

)). (D.0.10)

62

The other three contributions to the partition function may be similarly written down by combiningthe second, third and fourth terms respectively from A and B, we obtain the nested product forms

Z−1(2) =

∞∏k=0

∞∏m1,2,4

( ∞∏s=1

∞∏n=s

(1− (qq3)n qk+m1+m2+m4+2qm1

1 qm2+k2 qk3q

m4+k4

)), (D.0.11)

Z−1(3) =

∞∏k=0

∞∏m1,2,3

( ∞∏s=1

∞∏n=s

(1− (qq4)n qk+m1+m2+m4+2qm1+k

1 qm22 qm3+k

3 qk4

)), (D.0.12)

Z−1(4) =

∞∏k=0

∞∏m1,3,4

( ∞∏s=1

∞∏n=s

(1− (qq2)n qk+m1+m3+m4+2qm1+k

1 qk2qm33 qm4+k

4

)). (D.0.13)

Putting these expressions together yields the expression for Z in (3.3.28).

63

64

Appendix E

Classical analysis for generic spins

In this appendix, we will study the classical analysis for generic spins and show that k(s,m, µ`)(4.5.39) and r(s,m, µ`) ≡ k(s,m, µ`) + s −m will appear as weights (right and left respectively)of massive primaries in the linearised spectrum of generic spins.

The classical equation of motion for a generic spin is

FM1···Ms ≡ D(M)FM1M2···Ms ≡ FM1···Ms +1

s(s− 1)µεQR(M1

∇QFRM2···Ms)= 0, (E.0.1)

where

FM1M2M3.....Ms ≡ ∆φM1M2.....Ms −∇(M1|∇Qφ|M2.....Ms)Q +

1

2∇(M1

∇M2φP

M3M4....Ms)P

− 1

`2

[s2 − 3s

]φM1M2....Ms + 2g(M1M2

φ PM3M4...Ms)P

.

(E.0.2)

Using the field redefinition and the gauge condition defined below,

φM1···Ms = φM1···Ms −1

s2g(M1M2

φ(s−2)M3···Ms)

,

∇M φMM2···Ms =1

2∇(M2

φ(s−2)M3···Ms)

, (E.0.3)

we can writeFM1M2M3.....Ms = D(L)D(R)φM1···Ms , (E.0.4)

where

D(L)φM1···Ms ≡ φM1···Ms +`

s(s− 1)εQR(M1

∇QφRM2···Ms),

D(R)φM1···Ms ≡ φM1···Ms −`

s(s− 1)εQR(M1

∇QφRM2···Ms). (E.0.5)

One can now check that the equations of motion implies that

gM1M2∇M3 · · · ∇MsFM1···Ms = 0

=⇒ gM1M2∇M3 · · · ∇MsFM1···Ms = 0

=⇒ ∇M3 · · · ∇Ms φ(s−2)M3···Ms

= 0. (E.0.6)

Now let us write down the massive branch equation D(M)φ = 0, by operating on it from the leftby D(M) which is the same as D(M) with µ replaced by −µ. By replacing µ` = ±1 on the solutions

65

obtained here, we can recover the left and right branch solutions. The equations are

∆φM1···Ms −m2s

`2φM1···Ms =

(2s− 3)

2s2∇(M1

∇M2 φ(s−2)M3···Ms)

+1

`2s2(s2 − s+ 2)g(M1M2

φ(s−2)M3···Ms)

+1

s2g(M1M2

∆φ(s−2)M3···Ms)

− 1

s2g(M1M2

∇M3χ(s−3)M4···Ms)

,

∆φ(s−2)M3···Ms

−m2s−2

`2φ

(s−2)M3···Ms

=(2s− 5)

(s− 2)2∇(M3χ

(s−3)M4···Ms)

− 2

(s− 2)2 g(M3M4χ

(s−4)M5···Ms)

, s > 2

∆χ(s−3)M4···Ms

−m2s−3

`2χ

(s−3)M4···Ms

=(2s− 7)

(s− 3)2∇(M4χ

(s−4)M5···Ms)

− 2

(s− 3)2 g(M4M5χ

(s−5)M6···Ms)

, s > 3

...

∆χ(2)MN −

m22

`2χ

(2)MN =

3

4∇(Mχ

(1)N),

∆χ(1)M −

m21

`2χ

(1)M = 0, (E.0.7)

whereχ(s−m) = ∇M3 · · · ∇Mm φ

(s−2)M3···Ms

, (E.0.8)

and ms and ms−m are given in (4.5.35). The procedure to solve the above equations is same asthe spin-3 analysis done in [103]. We need to start with the last equation of (E.0.7), the solutionof which would be given by h − h = ±1, where h and h are the left and right weights. We willput the solution of χ(1) into the equation of χ(2) and decompose χ(2) into two parts, one carryingthe weights of χ(1) and the other would be transverse, the solution of which would be given byh − h = ±2. Following the steps similarly we will decompose χ(3) into three parts, one carryingthe weight of χ(1), another carrying the weights of transverse χ(2) and a part which is transverseon its own carrying weight h− h = ±3. In this way we will finally obtain the full solution. We arenot interested in obtaining the explicit form of the final solution (although there is no technicalobstacle in doing so). We will, however, be interested in obtaining the weights of the differentmodes present in the final solution. For that it is sufficient (as per the argument above) to analysethe transverse traceless parts of the equations (E.0.7). For that let us write the Laplacian actingon traceless rank-s tensor as

∆φM1···Ms =

[− 2

`2(L2 + L2

)− s(s+ 1)

`2

]φM1···Ms (E.0.9)

Where, L2 and L2 are the left and right SL(2, R) casimirs of the isometry group of AdS3 (see eq4.11 and 4.12 of [103]). The eigenvalues of L2 and L2 are given by h(1 − h) and h(1 − h), whereh and h are the left and right weights of the solution. We will also use the fact that for a rank-s,transverse traceless primary, h− h = ±s. Thus the different modes of equation (E.0.7), will carryweights h− h = ±s, h − h = ±(s − 2), h− h = ±(s − 3), · · · , h − h = ±1. By using (E.0.7) and(E.0.9), we obtain the weights of the different modes as

h− h = +s : h =s2 (µ`+ 1)− s (µ`− 1)

2s, h =

s(s− 1)(µ`− 1)

2s,

h− h = −s : h =s(s− 1)(µ`− 1)

2s, h =

s2 (µ`+ 1)− s (µ`− 1)

2s,

h− h = +(s−m) : h =(s−m+ 1)(s−m) + s(s− 1)µ`

2(s−m)= r(s,m, µ`),

h =s(s− 1)µ`− (s−m− 1)(s−m)

2(s−m)= k(s,m, µ`), (s− 1) ≥ m ≥ 2,

h− h = −(s−m) : h = k(s,m, µ`), h = r(s,m, µ`), (s− 1) ≥ m ≥ 2 (E.0.10)

66

The modes with negative h − h will not belong to the massive branch (they appear because wehad the operation of D(M) on the original equation of motion). But these modes will become theright branch solution by taking µ` = 1. The modes with positive h− h will belong to the massivebranch solution. For µ` = 1, they will become the left branch solutions. These are the weightsthat also appear in the one loop partition function at the chiral point (4.5.45).

67

68

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